#9160802

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

4558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=ghi,d^2=hi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067998);