#9730343

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

4235 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,e^2=gh,f^2=ghi,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, 9637539);