Finite Group Databasebeta

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C₂×#8442 #186560

order	512
nilpotency class	3
isoclinism	isoclinic class of #1619
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	#379
prime radical	C₂×#3482	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#186595, C₂×#186596, C₂×#186562, …
maximal normal subgroup of	C₂×#186595, C₂×#186596, C₂×#186562, …
maximal quotient of	#72239481, #72239486

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93756);