# How large can a symmetric generating set of a finite group be?

Let $$G$$ be a finite group of order $$n$$ and let $$\Delta$$ be its generating set. I’ll say that $$\Delta$$ generates $$G$$ symmetrically if for every permutation $$\pi$$ of $$\Delta$$ there exists $$f:G\rightarrow G$$ an automorphism of $$G$$ such that $$f\restriction\Delta=\pi$$.

How large can $$\Delta$$ be with respect to $$n$$? Specifically what’s the asymptotic behaviour? Is there a class of groups (along with their symmetric generating sets) of unbounded order such that $$n$$ is polynomially bounded by $$|\Delta|$$. Has any other research been done on these generating sets?