#9160808

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

4408 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,17)(13,21)(14,15)(16,20)(18,23)(22,24)(25,26,32,29)(27,33,37,31)(28,30,34,39)(35,40,38,36);(2,5)(6,7)(10,15)(12,22)(16,19)(18,21)(25,27,32,37)(26,33,29,31)(28,38,34,35)(30,36,39,40);(1,2)(3,6)(4,5)(7,8)(10,15)(11,20)(17,24)(18,21)(25,28,32,34)(26,30,29,39)(27,35,37,38)(31,36,33,40);(1,3)(2,6)(4,8)(5,7)(9,11)(10,16)(12,18)(13,17)(14,20)(15,19)(21,22)(23,24)(25,29,32,26)(27,31,37,33)(28,30,34,39)(35,40,38,36);(9,12,14,22)(10,17,15,24)(11,21,20,18)(13,16,23,19)(25,30)(26,34)(27,36)(28,29)(31,38)(32,39)(33,35)(37,40);(9,13,14,23)(10,18,15,21)(11,17,20,24)(12,19,22,16)(25,31)(26,27)(28,36)(29,37)(30,35)(32,33)(34,40)(38,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=ghi,d^2=ghi,e^2=h,f^2=h,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, 9068004);