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×#58145 |
| 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 | |
4924 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);(3,4,7,5)(6,9,10,8)(19,20)(21,23)(22,29)(24,26)(25,27)(28,31)(30,33)(32,34);(4,5)(8,9)(11,12)(13,14)(15,16)(17,18)(19,21,25,22)(20,23,27,29)(24,30,32,28)(26,33,34,31);(12,16)(13,17)(19,22,25,21)(20,23,27,29)(24,30,32,28)(26,31,34,33);(3,5,7,4)(6,9,10,8)(11,13,15,17)(12,14,16,18)(19,23)(20,22)(21,27)(24,31)(25,29)(26,30)(28,34)(32,33);(3,6,7,10)(4,8,5,9)(11,14)(12,17)(13,16)(15,18)(19,24,25,32)(20,26,27,34)(21,28,22,30)(23,31,29,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011715);