How many non-isomorphic finite groups with exactly $n$ conjugacy classes are there?
All I managed to prove was that their number is finite for all $n \in \mathbb{N}$:
Suppose $G$ has $n$ conjugacy classes $C_1, ... C_n$, and suppose $\forall i \leq n$ $g_i \in C_i$. Then $|C_i| = [G:C(g_i)]$. From that we can conclude, that $1 = |G|^{-1}(\sum_{i=1}^n |C_i|) = |G|^{-1}(\sum_{i=1}^n [G:C(g_i)])=\sum_{i=1}^n \frac{1}{C(g_i)}$. And it is known that there is only finitely many ways to partition $1$ into sum of $n$ Egyptian fractions.