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C₂×#21815 #6374290

order	512
nilpotency class	3
isoclinism	isoclinic class of #3186
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂²×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4904 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 18 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239585, #72239626, #72239630, …
maximal normal subgroup of	#72239585, #72239626, #72239630, …
maximal quotient of	#72239596

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281486);