#9232806

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
prime radical	#47108	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4446 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:

(1,2,4,6)(3,5,7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,23)(21,22)(25,26,31,29)(27,35,37,32)(28,38,36,33)(30,34,40,39);(3,7)(5,8)(11,18)(15,17)(19,20)(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)(9,11)(10,15)(12,19)(13,17)(14,18)(16,23)(20,21)(22,24)(25,26,31,29)(27,32,37,35)(28,33,36,38)(30,34,40,39);(9,12,14,21)(10,16,13,22)(11,19,18,20)(15,23,17,24)(25,28)(26,33)(27,30)(29,38)(31,36)(32,34)(35,39)(37,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,20)(13,15)(14,18)(16,23)(19,21)(22,24)(25,29,31,26)(27,35,37,32)(28,33,36,38)(30,34,40,39);(1,2)(3,5)(4,6)(7,8)(9,13,14,10)(11,17,18,15)(12,16,21,22)(19,23,20,24)(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=g,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, 9140002);