C₂×C₄×#125 #34034

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

1311 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 27022);