C₄×#341 #10704

order	256
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) order 131072
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

455 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2,3,4);(5,6)(7,8)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24);(6,8)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3692);