#9189446

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

4576 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=i,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9096642);