A faster method for computing conjugation classes I am asked to find the conjugation classes of a group order n. I am aware what a conjugation class is and how to find it. My question: is there a quicker/more simple way to find the conjugation classes of the group oppose to manually conjugating by each element of the group?
 A: One general thing you can do for finite groups is think about the class equation,
$$
|G|=|Z(G)|+\sum_x [G:C_G(x)],
$$
where the sum runs over representatives of each conjugacy class.
This in and of itself is not helpful; if you already knew all the conjugacy classes well enough to find representatives, you'd be done.  However, there are a few things we can garner from this that may be helpful.


*

*We do know $|G|$.

*We know that each of the $[G:C_G(x)]$ must be numbers that divide $|G|$, as is $|Z(G)|$ (which you can sometimes narrow down even further if, for example, the group is nilpotent). Depending on how big $|G|$ is, there may not be too many ways for it to be a sum of its divisors.  For most numbers small enough that you'd consider computing conjugacy classes by hand, you can write out all possibilities yourself.

*Every element of $G$ that is conjugate to $x$ has the same order as $x$. Therefore, if you know the orders of each element (or even just some of them), you can narrow down which elements even could be in the same classes, and therefore try to match partition them up into reasonable guesses before you do any checking.  In this way you can avoid checking a lot of conjugations in many cases.
These won't take you all the way.  In general, you'll still have to conjugate a lot of stuff around, but the cuts can make it shorter.
