Fixed-class tuple fraction

Definition
Suppose $$G$$ is a finite group and $$c$$ is a positive integer. Define the set:

$$CT_c(G) = \{ (x_1,x_2,\dots,x_{c+1}) \in G^{c+1} \mid [[ \dots [x_1,x_2],x_3],\dots,x_c],x_{c+1}] = e \}$$

Then, the class $$c$$ tuple fraction of $$G$$ is defined as:

$$\frac{|CT_c(G)|}{|G|^{c+1}}$$

Note that this fraction equals 1 if and only if $$G$$ is a nilpotent group of nilpotency class at most $$c$$.

Facts

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