Central product of D8 and Z8

Definition
This group can be described as the defining ingredient::central product of defining ingredient::dihedral group:D8 and defining ingredient::cyclic group:Z8 over a common cyclic central subgroup of order two.

Description by presentation
Here is the description by a presentation:

gap> F := FreeGroup(3);  gap> G := F/[F.1^8,F.2^2,F.3^2,F.1*F.2*F.1^(-1)*F.2^(-1)];  gap> G := F/[F.1^8,F.2^2,F.3^2,F.1*F.2*F.1^(-1)*F.2^(-1),F.3*F.1*F.3^(-1)*F.1^(-5),F.3*F.2*F.3^(-1)*F.1^4*F.2]; 