#6513323

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500006
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

2966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,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, 6420519);