Symmetric groups

Number of conjugacy classes
The number of conjugacy classes in the symmetric group equals the number of unordered integer partitions of the size of the underlying set. This can be seen in a more set-theoretic way as follows. Consider the cycle type map:

$$CT: Sym(S) \to Part(|S|)$$

that partitions the natural number $$n = |S|$$ into parts which equal the sizes of the cycles in the cycle decomposition of the given permutation. Then, the conjugacy classes in $$Sym(S)$$ are precisely the fibers of the map $$CT$$.

Thus, the conjugacy classes are indexed by the set of unordered integer partitions.

Sizes of conjugacy classes
The size of the centralizer of a conjugacy class with $$i_j$$ cycles of size $$j$$ is given by:

$$\prod_j (j)^{i_j} i_j!$$

In fact the centralizer is explicitly a direct product of groups $$D(j,i_j)$$ where each $$D(j,i_j)$$ is the semidirect product of $$(C_j)^{i_j}$$ with the symmetric group on $$i_j$$ elements.

Hence, the size of the conjugacy class is:

$$\frac{n!}{\prod_j (j)^{i_j} i_j!}$$

Degrees of irreducible representations
The irreducible representations correspond to the Young diagrams, which are again indexed by the set of unordered integer partitions. The degree of the irreducible representation equals the number of possible tableaux whose shape is that diagram, and this, by the hook formula, is:

$$\frac{n!}{Product of hook lengths}$$