#9160840

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

4452 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,e^2=i,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, 9068036);