Q₈×C₃₂ #13659

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×C₈×#1020 (order 3072) · Inn(G) = C₂² · Out(G) = C₂²×C₈×S₄
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂²

Aliases

Q₈×C₃₂

83 conjugacy classes of subgroups.

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8);(9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6647);