C₂×#3330 #61150

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×D₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1187 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);(3,4,6,8)(5,9,10,7)(12,16)(13,14)(17,18);(3,5)(4,7)(6,10)(8,9)(12,17)(16,18);(11,12)(13,16)(14,18)(15,17);(5,10)(7,9)(11,13,15,14)(12,16,17,18)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54138);