Finite Group Databasebeta

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C₂×#10927 #354071

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#354553, #72215962, #72215990, …
maximal normal subgroup of	C₂×#354553, #72215962, #72215990, …
maximal quotient of	C₂×#110284, C₂×#147427, #72319985, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261267);