#9788541

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
prime radical	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4234 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)(25,26,31,35)(27,36,39,32)(28,37,38,33)(29,30,34,40);(1,2,4,6)(3,5,8,7)(11,17)(15,21)(18,23)(22,24)(25,27,31,39)(26,32,35,36)(28,30,38,40)(29,33,34,37);(1,2)(3,5)(4,6)(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)(25,28,31,38)(26,33,35,37)(27,30,39,40)(29,32,34,36);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23)(25,29)(26,30)(27,37)(28,32)(31,34)(33,39)(35,40)(36,38);(1,3)(2,5)(4,8)(6,7)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(25,30)(26,34)(27,38)(28,39)(29,35)(31,40)(32,37)(33,36)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,d^2=g,f^2=h,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, 9695737);