Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(10,14)(15,17)(16,21)(22,23)(25,26)(27,36)(28,39)(29,40)(30,34)(31,32)(33,38)(35,37);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,27,32,38)(26,33,31,36)(28,37,30,29)(34,35,39,40);(2,5)(6,8)(12,20)(16,21)(18,24)(22,23)(25,28,32,30)(26,34,31,39)(27,37,38,29)(33,35,36,40);(3,7)(6,8)(9,10)(11,17)(12,16)(13,14)(15,19)(18,23)(20,21)(22,24)(25,29)(26,35)(27,28)(30,38)(31,40)(32,37)(33,39)(34,36);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,24)(21,23)(25,30,32,28)(26,34,31,39)(27,37,38,29)(33,40,36,35);(1,2)(3,6)(4,5)(7,8)(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,31,32,26)(27,36,38,33)(28,34,30,39)(29,40,37,35)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8806486);