#9779559

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

4763 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:

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

Polycyclic presentation:

2^9|b^2=g,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9686755);