#9133766

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

4533 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,e^2=g,f^2=g,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, 9040962);