#9729203

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

4365 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=i,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9636399);