#9224373

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

4555 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)(4,5)(6,8)(9,10,13,16)(11,14,19,17)(12,15,22,20)(18,23,21,24)(25,26,31,33)(27,32,36,40)(28,37,29,30)(34,35,38,39);(2,5)(6,7)(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,22)(21,24)(25,27,31,36)(26,32,33,40)(28,38,29,34)(30,35,37,39);(1,2)(3,6)(4,5)(7,8)(25,28,31,29)(26,30,33,37)(27,34,36,38)(32,39,40,35);(1,3,4,8)(2,6,5,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,23)(21,24)(25,29,31,28)(26,30,33,37)(27,34,36,38)(32,35,40,39);(9,11)(10,14)(12,21)(13,19)(15,24)(16,17)(18,22)(20,23)(25,27)(26,32)(28,34)(29,38)(30,39)(31,36)(33,40)(35,37);(9,12)(10,15)(11,18)(13,22)(14,23)(16,20)(17,24)(19,21)(25,30)(26,29)(27,35)(28,33)(31,37)(32,34)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9131569);