Fixed-class tuple fraction is bounded away from one for groups not of that class

Statement
Suppose $$c$$ is a positive integer. Suppose $$G$$ is a finite group that is not a nilpotent group of class at most $$c$$. In other words, $$G$$ may be a nilpotent group of class strictly greater than $$c$$ or it may be a non-nilpotent group.

Then, the class c tuple fraction of $$G$$ is at most:

$$1 - \frac{3}{2^{c + 2}}$$

Moreover, this bound is tight, because it is attained for any of the three maximal class groups of order $$2^{c+2}$$ (class $$c + 1$$) -- see classification of finite 2-groups of maximal class.

Related facts

 * Commuting fraction more than five-eighths implies abelian