Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#58042 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(19,20,25,28)(21,26,22,23)(24,27,32,34)(29,33,30,31);(4,7)(8,9)(11,12)(13,14)(15,16)(17,18)(19,21,25,22)(20,26,28,23)(24,30,32,29)(27,31,34,33);(12,16)(13,17)(19,22,25,21)(20,26,28,23)(24,30,32,29)(27,33,34,31);(3,4)(5,8)(6,7)(9,10)(11,13,15,17)(12,14,16,18)(19,23,25,26)(20,22,28,21)(24,31,32,33)(27,30,34,29);(3,5)(4,8)(6,10)(7,9)(11,14)(12,17)(13,16)(15,18)(19,24)(20,27)(21,29)(22,30)(23,31)(25,32)(26,33)(28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011706);