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C₂×#8286 #183446

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 512

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

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#229439, C₂×#183994, C₂×#183474, …
maximal normal subgroup of	C₂×#229439, C₂×#183994, C₂×#183474, …
maximal quotient of	C₂×#129173

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90642);