#9232824

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

4302 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)(9,10,15,19)(11,16,21,14)(12,17,22,23)(13,18,20,24)(25,26,31,29)(27,35,37,32)(28,38,36,33)(30,34,40,39);(3,7)(5,8)(9,11)(10,16)(12,13)(14,19)(15,21)(17,18)(20,22)(23,24)(25,27,31,37)(26,32,29,35)(28,30,36,40)(33,34,38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,20)(13,21)(14,18)(15,22)(16,24)(19,23)(25,26,31,29)(27,32,37,35)(28,33,36,38)(30,34,40,39);(9,13,15,20)(10,18,19,24)(11,12,21,22)(14,23,16,17)(25,28)(26,33)(27,30)(29,38)(31,36)(32,34)(35,39)(37,40);(1,3,4,7)(2,5,6,8)(9,12)(10,17)(11,13)(14,24)(15,22)(16,18)(19,23)(20,21)(25,29,31,26)(27,35,37,32)(28,33,36,38)(30,34,40,39);(1,2)(3,5)(4,6)(7,8)(9,14,15,16)(10,11,19,21)(12,18,22,24)(13,17,20,23)(25,30)(26,34)(27,36)(28,37)(29,39)(31,40)(32,38)(33,35)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=i,e^2=gh,f^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, 9140020);