#9160823

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

4550 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=hi,d^2=ghi,e^2=h,f^2=gh,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, 9068019);