Nilpotency class and order determine conjugacy class size statistics for groups up to prime-fourth order

Statement

Suppose $p$ is a prime number and $0\leq k\leq 4$. Then, for a group of order $p^{k}$, the Nilpotency class (?) $c$ of the group determines its conjugacy class size statistics.

Here is the complete list:

$k$ (prime-base logarithm of order)	$c$ (nilpotency class)	number of conjugacy classes of size 1	number of conjugacy classes of size $p$	number of conjugacy classes of size $p^{2}$
0	0	1	0	0
1	1	$p$	0	0
2	1	$p^{2}$	0	0
3	1	$p^{3}$	0	0
3	2	$p$	$p^{2}-1$	0
4	1	$p^{4}$	0	0
4	2	$p^{2}$	$p^{3}-p$	0
4	3	$p$	$p^{2}-1$	$p^{2}-p$

Related facts

Stronger facts

• Nilpotency class and order determine group up to commutator map-equivalence for up to prime-fourth order

Other similar facts

• Nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order

Opposite facts

• Nilpotency class and order need not determine conjugacy class size statistics for groups of prime-fifth order