#9726981

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

4324 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,26,30,31)(27,32,35,39)(28,33,38,37)(29,36,40,34);(1,3,6,7)(2,4,5,8)(11,17)(15,21)(18,23)(22,24)(26,31)(28,38)(32,39)(34,36);(1,2,6,5)(3,8,7,4)(10,14)(15,21)(16,19)(22,24)(26,31)(27,35)(33,37)(34,36);(1,4)(2,7)(3,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,27)(26,32)(28,36)(29,37)(30,35)(31,39)(33,40)(34,38);(1,4)(2,3)(5,7)(6,8)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28)(26,33)(27,36)(29,39)(30,38)(31,37)(32,40)(34,35);(1,5,6,2)(3,8,7,4)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,30,40)(26,34,31,36)(27,37,35,33)(28,32,38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=ghi,f^2=gi,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, 9634177);