#466738

order	512
nilpotency class	3
isoclinism	isoclinic class of #456400
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1341 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(6,7)(9,10,14,18)(11,19,20,32)(12,23,25,16)(13,26,27,17)(15,28,29,22)(21,35,34,33)(24,31,39,38)(30,36,37,40);(1,2,3,4)(5,6,8,7)(11,20)(15,29)(19,32)(21,34)(22,28)(30,37)(33,35)(36,40);(2,4)(5,6)(7,8)(9,11,13,22,14,20,27,28)(10,15,17,19,18,29,26,32)(12,21,24,36,25,34,39,40)(16,30,31,33,23,37,38,35);(5,7,8,6)(9,12,13,24,14,25,27,39)(10,16,17,31,18,23,26,38)(11,21,22,36,20,34,28,40)(15,30,19,33,29,37,32,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=ef,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373934);