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

This article gives a fact that is true for small groups of prime power order.More specifically, it is true for all groups of order $p^k$ where $k$ is at most equal to 4.
See more such facts| See more facts true for prime powers up to the same maximum power 4

Statement

Suppose $p$ is a prime number and $0 \le k \le 4$. 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