#6491464

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470037
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2892 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6398660);