#9222116

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

4615 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)(3,8)(4,6)(5,7)(9,10,15,18)(11,12,13,16)(14,22,24,17)(19,23,21,20)(25,26,31,32)(27,35,36,33)(28,38,37,34)(29,30,39,40);(1,3,5,4)(2,6,7,8)(9,11,15,13)(10,16,18,12)(14,19,24,21)(17,23,22,20)(26,32)(27,36)(29,39)(34,38);(9,12)(10,11)(13,18)(14,20)(15,16)(17,19)(21,22)(23,24)(25,27,31,36)(26,33,32,35)(28,29,37,39)(30,38,40,34);(1,2,5,7)(3,6,4,8)(9,13,15,11)(10,12,18,16)(14,19,24,21)(17,23,22,20)(25,28,31,37)(26,34,32,38)(27,29,36,39)(30,35,40,33);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,18)(14,23)(15,16)(17,19)(20,24)(21,22)(25,29)(26,30)(27,37)(28,36)(31,39)(32,40)(33,38)(34,35);(1,4,5,3)(2,6,7,8)(9,14)(10,17)(11,19)(12,20)(13,21)(15,24)(16,23)(18,22)(25,30)(26,29)(27,38)(28,35)(31,40)(32,39)(33,37)(34,36)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=g,f^2=gh,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, 9129312);