Also, just remembered that [G:Z(G)] = #G (order of G) when Z(G) is trivial, and if the index of the centralizer is #G, we have at most #G ways to conjugate any element of G.

Since Inn(G) ≈ G/Z(G) and Z(G) is trivial, we have Inn(G) ≈ G. Say ɸ is the homomorphism that maps G to Inn(G). Since we know this homomorphism is an isomorphism (bijective), does this mean that every element of G is conjugate to all elements of G?

Feel like I'm making progress, but still not sure what Aut(G) ≈ Inn(G) is telling us.