#9181804

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

4505 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=g,e^2=h,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, 9089000);