Finite Group Databasebeta

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C₂×#10836 #353619

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2462 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 128

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#411866, C₂×#369039, C₂×#393023, …
maximal normal subgroup of	C₂×#411866, C₂×#369039, C₂×#393023, …
maximal quotient of	C₂×#123426, C₂×#123854

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260815);