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

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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.

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