Orbital maximin equals size of set for nilpotent groups

Statement
Suppose $$S$$ is a set of size $$n$$. Consider the fact about::orbital maximin problem for nilpotent groups: find the maximum possible value of the size of the smallest orbital under the action of a nilpotent group on $$S$$. This maximum equals $$n$$. Note that the maximum is attained for the action of a cyclic group.

Related facts

 * Orbital maximin equals size of set for abelian groups
 * Orbital maximin equals size of set for faithful actions by groups with nontrivial center
 * Orbital maximin equals number of ordered pairs of distinct elements for solvable groups iff size is prime power
 * Orbital maximin is bounded below by constant fraction of number of ordered pairs of distinct elements for solvable groups