All partial sum values of squares of degrees of irreducible representations 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 partial sum values of squares of degrees of irreducible representations divide the order of the group.

In this case, it means that for any $$r \le k$$, the sum of squares of degrees of irreducible representations of $$P$$ that are at most $$p^r$$ is itself a power of $$p$$.

Opposite facts

 * There exist groups of prime-sixth order in which the partial sum values of squares of degrees of irreducible representations do not divide the order of the group

Related facts for conjugacy class sizes

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

 * Linear representation theory of groups of prime-cube order
 * Linear representation theory of groups of prime-fourth order
 * Linear representation theory of groups of prime-fifth order