Large orbits in actions of nilpotent groups

If a nontrivial nilpotent group $N$ acts faithfully and coprimely on a group $H$, it is shown that some element of $H$ has a small centralizer in $N$ and hence lies in a large orbit. Specifically, there exists $x \in H$ such that $|\mathbf{C}_{N}(x)| \le (|N|/p)^{1/p}$, where $p$ is the smallest prime divisor of $|N|$.