Order statistics of a finite group determine whether it is nilpotent

Statement
Whether or not a finite group is nilpotent is determined completely by its order statistics.

More specifically, consider a finite group of order $$n$$. This group is nilpotent if and only if, for every prime $$p$$ dividing the order of $$n$$, the number of elements of the group whose order is a power of $$p$$ is equal to the largest power of $$p$$ dividing $$n$$.

In particular, a finite nilpotent group cannot be order statistics-equivalent a finite group that is not nilpotent.