Finite Group Databasebeta

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C₂×#7938 #181105

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1920 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);(3,4,6,9,7,8,5,10)(11,12,14,18,16,17,15,13);(4,8)(5,6)(11,13,15,17,16,18,14,12);(3,5,7,6)(4,9,8,10)(12,17)(13,18)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#196204, C₂×#181106
maximal normal subgroup of	C₂×#196204, C₂×#181106
maximal quotient of	#72309185

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88301);