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



Automorphism Group of a p-group (finitely generated)
For a specific counterexample to the fact that $p^{2n}$ is not an upper bound, take the elementary abelian group of order $p^3$. It has automorphism group of order $(p^3-1)(p^3-p)(p^3-p^2)$ (pick a basis; the first basis vector can go to any nonzero vector; the second to any vector not in the linear span of the first; the third to any vector not in the linear span of the first two images). For $p=3$, this gives $11232$ automorphisms, larger than $(3^3)^2 = 729$. 


