#8533264

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

4476 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8440460);