#9159984

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

4409 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067180);