#9040227

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

4558 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)(25,26,31,34)(27,33,36,30)(28,32,29,39)(35,37,38,40);(1,3,4,7)(2,5,6,8)(9,10)(11,17)(12,20)(13,14)(15,19)(16,21)(18,23)(22,24)(25,27)(26,30)(28,35)(29,38)(31,36)(32,40)(33,34)(37,39);(1,2)(3,5)(4,6)(7,8)(10,14)(12,21)(15,17)(18,22)(25,28,31,29)(26,32,34,39)(27,35,36,38)(30,40,33,37);(9,11)(10,15)(12,22)(13,19)(14,17)(16,24)(18,21)(20,23)(25,29,31,28)(26,32,34,39)(27,35,36,38)(30,37,33,40);(1,3)(2,5)(4,7)(6,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,23)(17,24)(19,22)(25,30,31,33)(26,27,34,36)(28,37,29,40)(32,38,39,35);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,21)(22,24)(25,27)(26,33)(28,38)(29,35)(30,34)(31,36)(32,40)(37,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8947423);