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Apologies for yet another answer, but I really think I finally got it, as it fits all clues and your bonus hints!

My first semester—no, my entire freshman year—in high school, I was all alone.

In semester $n$ you are the identity in $A_n$. $A_1$ and $A_2$ don't have any other elements.

As I started my sophomore year, I finally made a friend. Two, actually.

Your new friends are the other two elements of $A_3$

They weren't in the same class as me, but for the rest of my high school life, they and their classmates basically defined my whole friend group.

They are not in your conjugacy class, but they and their classmates, together all 3-cycles (there are one or two conjugacy classes of 3-cycles, these two represent both classes), generate your whole friend group, namely $A_n$, which is generated by its 3-cycles.

As sophomore year drew to a close, I made yet more friends, from another class. There were finally some dynamics between us.

Next semester, more friends: $A_4$. Some from another class, namely that of the products of disjoint transpositions. Not sure about the dynamics.

Cue the start of junior year. More friends from yet another class, yet I feel things won't be so complicated anymore. (This semester we were big into D&D, but we had to substitute d12's for d20's).

Not so complicated anymore, namely simple: $A_n$ for $n \ge 5$ is simple. The new class(es) are the two classes of 5-cycles. $A_5$ is the symmetry group both of d12 and of d20.

At some point during my sixth semester, we looked at ourselves in the mirror and realized we were quite exceptional.

In the mirror you saw all your symmetries, those of $A_6$, which are exceptional: all other alternating groups have an outer automorphism group of (1 or) 2 elements, but that of $A_6$ is the Klein 4-group.

Start of senior year: Exponent? Blaze it!

The exponent of $A_7$ is 420. This is slang for smoking marihuana.

How many friends did I have by graduation, and across how many different classes?

You have $8!/2 - 1 = 20159$ friends, across 14 conjugacy classes, including your own.

Apologies for yet another answer, but I really think I finally got it, as it fits all clues and your bonus hints!

My first semester—no, my entire freshman year—in high school, I was all alone.

In semester $n$ you are the identity in $A_n$, the alternating group of even permutations of $n$ symbols. $A_1$ and $A_2$ don't have any other elements.

As I started my sophomore year, I finally made a friend. Two, actually.

Your new friends are the other two elements of $A_3$

They weren't in the same class as me, but for the rest of my high school life, they and their classmates basically defined my whole friend group.

They are not in your conjugacy class, but they and their classmates, together all 3-cycles (there are one or two conjugacy classes of 3-cycles, these two represent both classes), generate your whole friend group, namely $A_n$, which is generated by its 3-cycles.

As sophomore year drew to a close, I made yet more friends, from another class. There were finally some dynamics between us.

Next semester, more friends: $A_4$. Some from another class, namely that of the products of disjoint transpositions. Not sure about the dynamics.

Cue the start of junior year. More friends from yet another class, yet I feel things won't be so complicated anymore. (This semester we were big into D&D, but we had to substitute d12's for d20's).

Not so complicated anymore, namely simple: $A_n$ for $n \ge 5$ is simple. The new class(es) are the two classes of 5-cycles. $A_5$ is the symmetry group both of d12 and of d20.

At some point during my sixth semester, we looked at ourselves in the mirror and realized we were quite exceptional.

In the mirror you saw all your symmetries, those of $A_6$, which are exceptional: all other alternating groups have an outer automorphism group of (1 or) 2 elements, but that of $A_6$ is the Klein 4-group.

Start of senior year: Exponent? Blaze it!

The exponent of $A_7$ is 420. This is slang for smoking marihuana.

How many friends did I have by graduation, and across how many different classes?

You have $8!/2 - 1 = 20159$ friends, across 14 conjugacy classes, including your own.

Apologies for yet another answer, but I really think I finally got it, as it fits all clues and your bonus hints!

My first semester—no, my entire freshman year—in high school, I was all alone.

In semester $n$ you are the identity in $A_n$. $A_1$ and $A_2$ don't have any other elements.

As I started my sophomore year, I finally made a friend. Two, actually.

Your new friends are the other two elements of $A_3$

They weren't in the same class as me, but for the rest of my high school life, they and their classmates basically defined my whole friend group.

They are not in your conjugacy class, but they and their classmates, together all 3-cycles (there are one or two conjugacy classes of 3-cycles, these two represent both classes), generate your whole friend group, namely $A_n$, which is generated by its 3-cycles.

As sophomore year drew to a close, I made yet more friends, from another class. There were finally some dynamics between us.

Next semester, more friends: $A_4$. Some from another class, namely that of the products of disjoint transpositions. Not sure about the dynamics.

Cue the start of junior year. More friends from yet another class, yet I feel things won't be so complicated anymore. (This semester we were big into D&D, but we had to substitute d12's for d20's).

Not so complicated anymore, namely simple: $A_n$ for $n \ge 5$ is simple. The new class(es) are the two classes of 5-cycles. $A_5$ is the symmetry group both of d12 and of d20.

At some point during my sixth semester, we looked at ourselves in the mirror and realized we were quite exceptional.

In the mirror you saw all your symmetries, those of $A_6$, which are exceptional: all other alternating groups have an outer automorphism group of (1 or) 2 elements, but that of $A_6$ is the Klein 4-group.

Start of senior year: Exponent? Blaze it!

The exponent of $A_7$ is 420. This is slang for smoking marihuana.

How many friends did I have by graduation, and across how many different classes?

You have $8!/2 - 1 = 20159$ friends, across (if I counted correctly) 13 conjugacy classes, including your own.

Apologies for yet another answer, but I really think I finally got it, as it fits all clues and your bonus hints!

My first semester—no, my entire freshman year—in high school, I was all alone.

In semester $n$ you are the identity in $A_n$. $A_1$ and $A_2$ don't have any other elements.

As I started my sophomore year, I finally made a friend. Two, actually.

Your new friends are the other two elements of $A_3$

They weren't in the same class as me, but for the rest of my high school life, they and their classmates basically defined my whole friend group.

They are not in your conjugacy class, but they and their classmates, together all 3-cycles (there are one or two conjugacy classes of 3-cycles, these two represent both classes), generate your whole friend group, namely $A_n$, which is generated by its 3-cycles.

As sophomore year drew to a close, I made yet more friends, from another class. There were finally some dynamics between us.

Next semester, more friends: $A_4$. Some from another class, namely that of the products of disjoint transpositions. Not sure about the dynamics.

Cue the start of junior year. More friends from yet another class, yet I feel things won't be so complicated anymore. (This semester we were big into D&D, but we had to substitute d12's for d20's).

Not so complicated anymore, namely simple: $A_n$ for $n \ge 5$ is simple. The new class(es) are the two classes of 5-cycles. $A_5$ is the symmetry group both of d12 and of d20.

At some point during my sixth semester, we looked at ourselves in the mirror and realized we were quite exceptional.

In the mirror you saw all your symmetries, those of $A_6$, which are exceptional: all other alternating groups have an outer automorphism group of (1 or) 2 elements, but that of $A_6$ is the Klein 4-group.

Start of senior year: Exponent? Blaze it!

The exponent of $A_7$ is 420. This is slang for smoking marihuana.

How many friends did I have by graduation, and across how many different classes?

You have $8!/2 - 1 = 20159$ friends, across 14 conjugacy classes, including your own.

Apologies for yet another answer, but I really think I finally got it, as it fits all clues and your bonus hints!

My first semester—no, my entire freshman year—in high school, I was all alone.

In semester $n$ you are the identity in $A_n$. $A_1$ and $A_2$ don't have any other elements.

As I started my sophomore year, I finally made a friend. Two, actually.

Your new friends are the other two elements of $A_3$

They weren't in the same class as me, but for the rest of my high school life, they and their classmates basically defined my whole friend group.

They are not in your conjugacy class, but they and their classmates, together all 3-cycles (there are one or two conjugacy classes of 3-cycles, these two represent both classes), generate your whole friend group, namely $A_n$, which is generated by its 3-cycles.

As sophomore year drew to a close, I made yet more friends, from another class. There were finally some dynamics between us.

Next semester, more friends: $A_4$. Some from another class, namely that of the products of disjoint transpositions. Not sure about the dynamics.

Cue the start of junior year. More friends from yet another class, yet I feel things won't be so complicated anymore. (This semester we were big into D&D, but we had to substitute d12's for d20's).

Not so complicated anymore, namely simple: $A_n$ for $n \ge 5$ is simple. $A_5$ is the symmetry group both of d12 and of d20.

At some point during my sixth semester, we looked at ourselves in the mirror and realized we were quite exceptional.

In the mirror you saw all your symmetries, those of $A_6$, which are exceptional: all other alternating groups have an outer automorphism group of 2 elements, but that of $A_6$ is the Klein 4-group.

Start of senior year: Exponent? Blaze it!

The exponent of $A_7$ is 420. This is slang for smoking marihuana.

How many friends did I have by graduation, and across how many different classes?

You have $8!/2 - 1 = 20159$ friends, across (if I counted correctly) 13 conjugacy classes, including your own.

Apologies for yet another answer, but I really think I finally got it, as it fits all clues and your bonus hints!

My first semester—no, my entire freshman year—in high school, I was all alone.

In semester $n$ you are the identity in $A_n$. $A_1$ and $A_2$ don't have any other elements.

As I started my sophomore year, I finally made a friend. Two, actually.

Your new friends are the other two elements of $A_3$

They weren't in the same class as me, but for the rest of my high school life, they and their classmates basically defined my whole friend group.

They are not in your conjugacy class, but they and their classmates, together all 3-cycles (there are one or two conjugacy classes of 3-cycles, these two represent both classes), generate your whole friend group, namely $A_n$, which is generated by its 3-cycles.

As sophomore year drew to a close, I made yet more friends, from another class. There were finally some dynamics between us.

Next semester, more friends: $A_4$. Some from another class, namely that of the products of disjoint transpositions. Not sure about the dynamics.

Cue the start of junior year. More friends from yet another class, yet I feel things won't be so complicated anymore. (This semester we were big into D&D, but we had to substitute d12's for d20's).

Not so complicated anymore, namely simple: $A_n$ for $n \ge 5$ is simple. The new class(es) are the two classes of 5-cycles. $A_5$ is the symmetry group both of d12 and of d20.

At some point during my sixth semester, we looked at ourselves in the mirror and realized we were quite exceptional.

In the mirror you saw all your symmetries, those of $A_6$, which are exceptional: all other alternating groups have an outer automorphism group of (1 or) 2 elements, but that of $A_6$ is the Klein 4-group.

Start of senior year: Exponent? Blaze it!

The exponent of $A_7$ is 420. This is slang for smoking marihuana.

How many friends did I have by graduation, and across how many different classes?

You have $8!/2 - 1 = 20159$ friends, across (if I counted correctly) 13 conjugacy classes, including your own.