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

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.