SmallGroup(32,27)

Definition
This group is a semidirect product of elementary abelian group:E8 and Klein four-group where the latter acts faithfully by transvections relative to a particular plane. It is given by the following presentation:

$$\langle x,y,z,a,b \mid x^2 = y^2 = z^2 = a^2 = b^2 = e, xy = yx, xz, = zx, yz = zy, aza^{-1} = xz, bzb^{-1} = yz, ax = xa, ay = ya, bx = xb, by = yb \rangle$$

It can also be described as the subgroup of upper-triangular unipotent matrix group:U(4,2) given by matrices with the $$(1,2)$$-entry equal to zero, i.e., matrices of the form:

$$\begin{pmatrix} 1 & 0 & * & * \\ 0 & 1 & * & * \\ 0 & 0 & 1 & * \\ 0 & 0 & 0 & 1 \\\end{pmatrix}$$

It can also be defined as the 2-Sylow subgroup of the automorphism group of the homocyclic group given as the direct product of Z4 and Z4.

Another group that occurs as a faithful semidirect product of the elementary abelian group of order eight and the Klein four-group is SmallGroup(32,49).