D₄×C₃₂ #13644

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂²×C₈×#120 (order 1024) · Inn(G) = C₂² · Out(G) = C₂²×C₈×D₄
homology	Abelianization (H₁) = C₂²×C₃₂ · Schur multiplier (H₂) = C₂³

Aliases

D₄×C₃₂

107 conjugacy classes of subgroups.

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6632);