#8429519

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

4411 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)(10,14)(15,21)(16,19)(22,23)(26,32)(27,29)(28,36)(39,40);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,22)(18,21)(19,20)(23,24)(25,26)(27,30)(28,37)(29,34)(31,32)(33,36)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(25,27,31,29)(26,30,32,34)(28,35,36,38)(33,40,37,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,23)(18,21)(22,24)(25,28)(26,33)(27,35)(29,38)(30,40)(31,36)(32,37)(34,39);(1,3,4,7)(2,6,5,8)(9,11,13,18)(10,15,14,21)(12,17,20,24)(16,22,19,23)(25,29,31,27)(26,34,32,30)(28,38,36,35)(33,39,37,40);(9,12)(10,16)(11,17)(13,20)(14,19)(15,22)(18,24)(21,23)(25,30,31,34)(26,27,32,29)(28,39,36,40)(33,38,37,35)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8336715);