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

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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

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