C₂×#3334 #61225

order	256
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54213);