#9161222

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

4634 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9068418);