Q₈×#124 #36548

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1163 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29536);