#9224360

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

4544 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)(3,7,8,5)(9,10)(11,16)(12,22)(13,14)(15,18)(17,21)(19,24)(20,23)(25,26)(27,36)(28,39)(29,40)(30,34)(31,32)(33,38)(35,37);(1,3,6,8)(2,5,4,7)(9,11)(10,16)(12,23)(13,19)(14,24)(15,21)(17,18)(20,22)(25,27,32,38)(26,33,31,36)(28,37,30,29)(34,35,39,40);(9,12,15,13)(10,14,18,22)(11,19,21,23)(16,20,17,24)(25,28,32,30)(26,34,31,39)(27,37,38,29)(33,35,36,40);(2,4)(5,7)(9,13,15,12)(10,14,18,22)(11,19,21,23)(16,24,17,20)(25,29)(26,35)(27,28)(30,38)(31,40)(32,37)(33,39)(34,36);(1,4,6,2)(3,7,8,5)(9,14)(10,13)(11,20)(12,18)(15,22)(16,23)(17,19)(21,24)(25,30,32,28)(26,34,31,39)(27,37,38,29)(33,40,36,35);(1,5,6,7)(2,3,4,8)(9,11)(10,17)(12,19)(13,23)(14,24)(15,21)(16,18)(20,22)(25,31,32,26)(27,36,38,33)(28,39,30,34)(29,35,37,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=i,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9131556);