C₂×#3194 #60737

order	256
nilpotency class	3
isoclinism	isoclinic class of #3192
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 ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1343 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

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

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53725);