#9726559

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9633755);