SmallGroup(256,6745)

Definition
This group of order 256 is defined by means of the following presentation (here $$e$$ denote the identity element):

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

Here $$[ \, \ ]$$ denotes the commutator of two elements. Because the group has nilpotency class two, the choice of convention for commutator (left or right) does not matter.

Description by presentation
gap> F := FreeGroup(4);  gap> G := F/[F.1^4,F.2^4,F.3^4,F.4^4,Comm(F.1,F.2)*F.3^(-2),Comm(F.1,F.3),Comm(F.2,F.3)*F.4^(-2),Comm(F.1,F.4),Comm(F.2,F.4),Comm(F.3,F.4)];  gap> IdGroup(G); [ 256, 6745 ]