SmallGroup(256,29626)

Definition
This group of order 256 is defined by the following presentation:

$$\! G := \langle a_1, a_2, a_3, a_4, a_5 \mid a_1^2 = a_2^2 = a_3^4 = a_4^4 = a_5^4 = e, [a_1,a_2] = a_3^2, [a_1,a_3] = e, [a_2,a_3] = a_4^2, [a_1,a_4] = [a_3,a_4] = e, [a_2,a_4] = a_5^2, [a_1,a_5] = [a_2,a_5] = [a_3,a_5] = [a_4,a_5] = e \rangle$$

Description by presentation
gap> F := FreeGroup(5);  gap> R1 := [F.1^2,F.2^2,F.3^4,F.4^4,F.5^4]; [ f1^2, f2^2, f3^4, f4^4, f5^4 ] gap> R2 := [Comm(F.1,F.2)*F.3^(-2)]; [ f1^-1*f2^-1*f1*f2*f3^-2 ] gap> R3 := [Comm(F.1,F.3),Comm(F.2,F.3)*F.4^(-2)]; [ f1^-1*f3^-1*f1*f3, f2^-1*f3^-1*f2*f3*f4^-2 ] gap> R4 := [Comm(F.1,F.4),Comm(F.3,F.4),Comm(F.2,F.4)*F.5^(-2)]; [ f1^-1*f4^-1*f1*f4, f3^-1*f4^-1*f3*f4, f2^-1*f4^-1*f2*f4*f5^-2 ] gap> R5 := [Comm(F.1,F.5),Comm(F.2,F.5),Comm(F.3,F.5),Comm(F.4,F.5)]; [ f1^-1*f5^-1*f1*f5, f2^-1*f5^-1*f2*f5, f3^-1*f5^-1*f3*f5, f4^-1*f5^-1*f4*f5 ] gap> R := Union([R1,R2,R3,R4,R5]); [ f1^2, f2^2, f1^-1*f3^-1*f1*f3, f1^-1*f4^-1*f1*f4, f1^-1*f5^-1*f1*f5, f2^-1*f5^-1*f2*f5, f3^-1*f4^-1*f3*f4, f3^-1*f5^-1*f3*f5, f3^4, f4^-1*f5^-1*f4*f5, f4^4, f5^4, f1^-1*f2^-1*f1*f2*f3^-2, f2^-1*f3^-1*f2*f3*f4^-2, f2^-1*f4^-1*f2*f4*f5^-2 ] gap> G := F/R;  gap> Order(G); 256 gap> IdGroup(G); [ 256, 29626 ]