All cumulative conjugacy class size statistics values divide the order of the group for groups up to prime-fifth order

Statement
Suppose $$p$$ is a prime number and $$k$$ is an integer satisfying $$0 \le k \le 5$$. Suppose $$P$$ is a group of order $$p^k$$. Then, $$P$$ is a proves property satisfaction of::finite group in which all cumulative conjugacy class size statistics values divide the order of the group.

In this case, it means that for any $$r \le k$$, the number of elements of $$P$$ whose conjugacy class has size dividing $$p^r$$ is itself a power of $$p$$.

Opposite facts

 * There exist groups of prime-sixth order in which the cumulative conjugacy class size statistics values do not divide the order of the group

Related specific information

 * Element structure of groups of prime-cube order
 * Element structure of groups of prime-fourth order
 * Element structure of groups of prime-fifth order