Conjugacy class size statistics need not determine degrees of irreducible representations

Statement
It is possible to have two finite groups $$G_1$$ and $$G_2$$ such that the conjugacy class size statistics of $$G_1$$ are the same as those of $$G_2$$ (i.e., the two groups have the same number of conjugacy classes of each size) but the fact about::degrees of irreducible representations over $$\mathbb{C}$$ for $$G_1$$ are not the same as those of $$G_2$$.

Converse

 * Degrees of irreducible representations need not determine conjugacy class size statistics

Similar facts

 * Conjugacy class size statistics need not determine group up to isoclinism
 * Conjugacy class size statistics need not determine nilpotency class
 * Conjugacy class size statistics need not determine derived length

Proof
The smallest examples are among groups of order 128.