#9726583

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

4359 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=gh,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9633779);