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×#57901 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4750 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)(6,10)(7,8)(11,12,16,14)(13,17,21,19)(15,22,23,18)(20,25,26,24)(27,28)(29,32)(30,38)(31,34)(33,39)(35,37)(36,41)(40,42);(4,8)(5,10)(12,14)(17,19)(18,22)(24,25)(27,28,31,34)(29,35,37,32)(30,38,39,33)(36,40,42,41);(3,5)(4,6)(7,10)(8,9)(11,13,16,21)(12,17,14,19)(15,20,23,26)(18,24,22,25)(27,28)(29,35)(30,33)(31,34)(32,37)(36,41)(38,39)(40,42);(3,5)(4,6)(7,10)(8,9)(11,14,16,12)(13,19,21,17)(15,22,23,18)(20,25,26,24)(27,29,31,37)(28,32,34,35)(30,36,39,42)(33,40,38,41);(3,6)(4,5)(7,9)(8,10)(11,15,16,23)(12,18,14,22)(13,20,21,26)(17,24,19,25)(27,30,31,39)(28,33,34,38)(29,36,37,42)(32,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008500);