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C₂×#8967 #193868

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	#386
prime radical	C₂²×#120	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2912 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)(5,9,10,7)(6,8)(11,12,15,17,14,13,16,18);(3,4)(5,7)(6,8)(9,10)(11,13,15,18,14,12,16,17);(3,5)(4,7)(6,10)(8,9)(12,13)(17,18)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#193892, C₂×#248551, #72240595, …
maximal normal subgroup of	C₂×#193892, C₂×#248551, #72240595, …
maximal quotient of	#72264353, #72278640, #72303516, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101064);