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C₂×#20667 #6370819

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁶	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5090 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 22 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#6442584, #72237187, #72237189, …
maximal normal subgroup of	C₂×#6442584, #72237187, #72237189, …
maximal quotient of	#72239131, D₃×#20667, #72305485

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278015);