#9017550

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

4516 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,22)(21,24)(25,26)(27,34)(28,33)(29,32)(30,35)(31,36)(37,40)(38,39);(2,5)(6,7)(9,11,14,17)(10,15,13,19)(12,18,20,21)(16,23,22,24)(25,27,31,29)(26,32,36,34)(28,37,39,30)(33,35,38,40);(1,2)(3,6)(4,5)(7,8)(10,13)(11,17)(12,20)(23,24)(25,28,31,39)(26,33,36,38)(27,37,29,30)(32,35,34,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,16)(11,18)(13,22)(14,20)(15,23)(17,21)(19,24)(25,29,31,27)(26,34,36,32)(28,37,39,30)(33,35,38,40);(9,11)(10,15)(12,21)(13,19)(14,17)(16,24)(18,20)(22,23)(25,30,31,37)(26,35,36,40)(27,28,29,39)(32,38,34,33);(9,13,14,10)(11,19,17,15)(12,22,20,16)(18,24,21,23)(25,26)(27,32)(28,38)(29,34)(30,35)(31,36)(33,39)(37,40)

Polycyclic presentation:

2^9|b^2=gi,c^2=ghi,d^2=ghi,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924746);