#9164008

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

4524 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=g,e^2=ghi,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, 9071204);