Does someone know whether the order of the automorphism group of a general p-group of order $p^n$ is bounded from above by $(p^n)^2 $? (Every element can possibly be transferred to one of other $p^n$ elements)...

If this fact is incorrect, is it possible to deduce a bound on the order of such an automorphism group if our p-group is finitely generated?

Thanks in advance

`$O(p^{2n})$`

but`$O(p^{n^2})$`

. – Andreas Blass May 30 '12 at 17:12