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 $G$. However, that number is a multiple of $|\text{Aut}(G)|$. Indeed, if $\cal B$ is the set of all bases of $G$, then all $\text{Aut}(G)$-orbits on $\cal B$ have the same cardinality $|\text{Aut}(G)|$.