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C₂×#9472 #201375

order	512
nilpotency class	3
isoclinism	isoclinic class of #9471
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2408	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 18 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#201386, C₂×#248551, #72278664, …
maximal normal subgroup of	C₂×#201386, C₂×#248551, #72278664, …
maximal quotient of	#72264362, #72282575, #72313926

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108571);