Show that $|Aut(G)|<n^{\log_2(n)}$ where $G$ is finite

Question

Let $G$ be a finite group of order $n>1$. Show that $|Aut($G$)|<n^{\log_2(n)}$.

The 'only' thing I know is the group of inner automorphisms is isomorphic to $G/Z(G)$ and by definition, $Z(G)=G$ if and only if $G$ is abelian.

If $|G|>2$ this thread show that Aut($G$) contains at least two elements.

But how can I have an upper bound ?

Answer 1

Here are some hints - more details on request.

1.Show first that any finite group of order $n$ has a generating set $X$ with $|X| \le \log_2(n)$.

2.Show that any homomorphism $\phi$ with domain $G$ is completely determined by the images of $\phi$ on a generating set $X$.

3.Deduce the result, taking care to prove the strict inequality.