[GAP Forum] automorphisms of "large" 2-groups

[Benjamin]

Dear GAP users,

I need some help with the following task: Consider

P:=SmallGroup(2^9,10477010);

This group satisfies Z(P)=Phi(P)=Omega(P) and Z(P) has order 8. All I 
want to know is if Aut(P) is a 2-group. The commands 
AutomorphismGroup(P) and AutomorphismGroupPGroup(P) (using the AutPGrp 
package) seem to take very long (have been running for hours). Therefore 
I guess Aut(P) is quite big and definitely not a 2-group. On the other 
hand, I tried to extend automorphisms of odd order of subgroups and 
quotient groups without success. In fact, I believe I showed that any 
nontrivial automorphism of odd order must have order 7 (with regular 
action on Z(P)).

In any case it would be nice to write down a nontrivial automorphism of 
odd order without knowing them all. Otherwise I would appreciate any 
argument that Aut(P) is in fact a 2-group.

Many thanks,
Benjamin