#9006659

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

4510 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)(3,8)(4,7)(5,6)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,31,34)(27,32,29,30)(28,33,36,40)(35,39,38,37);(3,4)(7,8)(10,14)(15,17)(16,20)(22,23)(25,27,31,29)(26,32,34,30)(28,35,36,38)(33,39,40,37);(1,3)(2,7)(4,6)(5,8)(10,14)(11,18)(12,21)(22,23)(27,29)(28,36)(30,32)(33,40);(3,4)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,36)(32,39)(34,40);(1,4,6,3)(2,7,5,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,29,31,27)(26,30,34,32)(28,35,36,38)(33,39,40,37);(1,5,6,2)(3,7,4,8)(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,30)(26,27)(28,37)(29,34)(31,32)(33,35)(36,39)(38,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,e^2=i,f^2=gh,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, 8913855);