C₂×#3386 #61360

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
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1003 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);(3,4,7,6)(5,9,10,8)(11,12,15,18)(13,16,14,17);(3,5,7,10)(4,8,6,9)(11,13,15,14)(12,16,18,17);(12,17)(13,14)(16,18);(3,6,7,4)(5,8,10,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54348);