C₂×C₄×#137 #60157

order	256
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

1405 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 22 points:

(1,2);(3,4,5,6);(7,8)(9,15)(10,18)(11,13)(12,16)(14,20)(17,22)(19,21);(7,9,11,17)(8,12,13,21)(10,19,20,16)(14,22,18,15);(8,13)(12,21)(14,18)(15,22)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53145);