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×#54355 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4468 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);(3,4)(5,9)(6,8)(7,10)(11,12,16,14)(13,17,21,19)(15,22,23,18)(20,25,26,24)(27,28)(29,33)(30,34)(31,32)(35,37)(36,41)(38,40)(39,42);(3,5,8,7)(4,9,6,10)(12,14)(17,19)(18,22)(24,25)(28,31)(29,35)(30,38)(41,42);(3,6,8,4)(5,9,7,10)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,29,32,35)(28,33,31,37)(30,39,38,36)(34,42,40,41);(3,7,8,5)(4,9,6,10)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,30,32,38)(28,34,31,40)(29,36,35,39)(33,41,37,42);(11,15,16,23)(12,18,14,22)(13,20,21,26)(17,24,19,25)(27,31,32,28)(29,37,35,33)(30,40,38,34)(36,42,39,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011048);