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)
Y.B.
Yakov
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