Central product of M16 and Z8 over common Z2

Definition
This group is defined as the central product of the groups defining ingredient::M16 and defining ingredient::cyclic group:Z8 over a common cyclic central subgroup cyclic group:Z2. Explicitly, it has the presentation:

$$G := \langle a,b,c \mid a^8 = b^2 = c^8 = e, a^4 = c^4, ac = ca, bc = cb, bab^{-1} = a^5 \rangle$$

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