In fact, if $|G|=p^n$ and $d(G)=d$, then
$|\mathrm{Aut}(G)|$ divides the number $$(p^n-p^{n-d})(p^n-p^{n-d+1}) ... (p^n-p^{n-1}).$$ This result is due to P. Hall (1933)

Proof. The above product is the number of minimal bases of $G$. Howver, that number is a multiple of $|\text{Aut}(G)|$.