SmallGroup(128,1015)

Definition
This is a group of order 128 given by the following presentation, where $$e$$ is used to denote the identity element:

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

For the definition, $$[, ]$$ stands for the commutator. It does not matter whether we choose the left or right action convention for the commutator -- the groups defined in both cases are isomorphic.

Description by presentation
gap> F := FreeGroup(4);  gap> G := F/[F.1^2,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); [ 128, 1015 ]