Finite Group Databasebeta

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C₂×#11080 #357745

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	#385
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 22 points:

(1,2);(3,4)(5,6)(7,8,11,14,12,15,10,18)(9,13,16,22,17,21,19,20);(9,16,17,19)(13,20,21,22);(4,6)(7,9)(8,13)(10,19)(11,16)(12,17)(14,22)(15,21)(18,20)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214944, #72215689, C₂×#379089, …
maximal normal subgroup of	#72214944, #72215689, C₂×#379089, …
maximal quotient of	#72214917, #72215245, #72215279, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264941);