#9668320

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

4757 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9575516);