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×#58341 |
| 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 | |
4582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(11,12)(13,17)(14,21)(15,16)(18,22)(19,25)(20,23)(24,26)(27,28,32,31)(29,35,36,33)(30,38,39,34)(37,40,42,41);(3,4,7,5)(6,9,10,8)(12,16)(17,23)(18,21)(24,25)(29,36)(33,35)(37,42)(40,41);(4,5)(8,9)(11,13,15,20)(12,17,16,23)(14,19,22,26)(18,24,21,25)(27,29)(28,33)(30,37)(31,35)(32,36)(34,40)(38,41)(39,42);(3,5,7,4)(6,9,10,8)(11,12)(13,17)(14,18)(15,16)(19,24)(20,23)(21,22)(25,26)(27,30,32,39)(28,34,31,38)(29,37,36,42)(33,40,35,41);(3,6)(4,8)(5,9)(7,10)(11,14)(12,18)(13,19)(15,22)(16,21)(17,24)(20,26)(23,25)(27,31,32,28)(29,35,36,33)(30,38,39,34)(37,41,42,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011289);