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.