Thank you for several replies!
Heiko Dietrich pointed out that the computation can be done with Magma
in a matter of seconds. Also Eamonn O'Brien provided a concrete
automorphism of order 7.
I like to add that I had no problems with GAP computing automorphism
groups of very similar groups (of order 2^9).
Best wishes,
Benjamin
Am 10.02.2014 15:55, schrieb 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
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