For the symmetric groups $S_n$ there's a quite simple way to find all the conjugacy classes. We find partitions of $n$, that is, we write
$$n = n_1+\cdots+n_k,$$
and then for each such partition there's a conjugacy class which is the set of all permutations of the form
$$\sigma=\sigma_1\cdots\sigma_k,$$
being $\sigma_i$ one $n_i$-cycle.
Now, what about $A_n$, the alternating groups? Is there any systematic way like that to find all the conjugacy classes?
