Say we have a finite group $G$ generated by $\langle g_1,g_2,\ldots ,g_n\rangle$ which is a minimal generating set. What information is required from this set in order to determine the whole group?

For cyclic groups, all that is needed is the order of some $g_1$, but for a set generated by two elements, I imagine you need the order of $g_1$, $g_2$, and the order of some combination of the two elements.

The motivation is to better understand Cayley graphs and potentially use these ideas to better understand group extensions.