#9221208

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

4608 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,d^2=h,e^2=g,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, 9128404);