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C₂×#9255 #198372

order	512
nilpotency class	3
isoclinism	isoclinic class of #9239
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231679, C₂×#198404, C₂×#248456, …
maximal normal subgroup of	#72231679, C₂×#198404, C₂×#248456, …
maximal quotient of	#72264321, #72264333, #72282264, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105568);