#9210525

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

4400 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)(25,26)(27,36)(28,33)(29,30)(31,34)(32,35)(37,39)(38,40);(3,7)(5,8)(9,10)(11,15)(12,16)(13,17)(14,19)(18,24)(20,23)(21,22)(25,27)(26,31)(28,37)(29,38)(30,39)(32,34)(33,40)(35,36);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,23)(20,22)(21,24)(25,28,32,29)(26,33,35,30)(27,37,34,38)(31,40,36,39);(9,11)(10,15)(12,20)(13,19)(14,17)(16,23)(18,22)(21,24)(25,29,32,28)(26,33,35,30)(27,38,34,37)(31,40,36,39);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,21)(13,17)(15,19)(16,22)(18,24)(20,23)(25,30,32,33)(26,29,35,28)(27,39,34,40)(31,38,36,37);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,22)(14,23)(15,24)(17,21)(19,20)(25,31,32,36)(26,34,35,27)(28,40,29,39)(30,37,33,38)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=g,f^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, 9117721);