#6631415

order	512
nilpotency class	3
isoclinism	isoclinic class of #6631116
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^c=dfh,g^c=gi,e^d=ei

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, 6538611);