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?