C₂×#3397 #61394

order	256
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

	subgroup	G/H
center, socle	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1125 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54382);