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×#57686 |
| 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 | |
4652 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);(3,4,6,7)(5,9,10,8)(11,12,16,19)(13,14,21,17)(15,18,23,25)(20,22,26,24)(27,28,31,35)(29,33,37,32)(30,34,39,42)(36,41,38,40);(4,7)(5,10)(15,23)(18,25)(20,26)(22,24)(27,29,31,37)(28,32,35,33)(30,38,39,36)(34,41,42,40);(4,7)(8,9)(11,13,16,21)(12,17,19,14)(15,20,23,26)(18,24,25,22)(27,29)(28,33)(30,38)(31,37)(32,35)(34,40)(36,39)(41,42);(3,4)(5,8)(6,7)(9,10)(11,14,16,17)(12,13,19,21)(15,22,23,24)(18,20,25,26)(27,30,31,39)(28,34,35,42)(29,36,37,38)(32,40,33,41);(3,5)(4,8)(6,10)(7,9)(11,15)(12,18)(13,20)(14,22)(16,23)(17,24)(19,25)(21,26)(27,28)(29,33)(30,34)(31,35)(32,37)(36,41)(38,40)(39,42)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008597);