Orbital maximin equals size of set for faithful actions by groups with nontrivial center

Statement
Suppose $$S$$ is a set of size $$n$$. Consider the fact about::orbital maximin problem for faithful group actions by groups with a nontrivial center. In other words, consider the maximum possible value of the size of the smallest orbital under a faithful group action on $$S$$ by a group with nontrivial center. This maximum equals $$n$$. (The maximum is attained by a cyclic group acting on $$n$$).

Related facts

 * Orbital maximin equals size of set for abelian groups
 * Orbital maximax equals size of set for abelian groups
 * Orbital maximin equals size of set for nilpotent groups
 * 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