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×#54610 |
| 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 | |
4522 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)(5,7)(6,8)(9,10)(11,12,16,15)(13,19,20,17)(14,22,23,18)(21,24,26,25)(27,28,31,35)(29,33,37,32)(30,34,39,42)(36,41,38,40);(6,10)(8,9)(13,20)(17,19)(21,26)(24,25)(27,29,31,37)(28,32,35,33)(30,38,39,36)(34,41,42,40);(4,5)(8,9)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,25)(23,26)(27,29)(28,33)(30,38)(31,37)(32,35)(34,40)(36,39)(41,42);(3,5,7,4)(6,9,10,8)(11,14,16,23)(12,18,15,22)(13,21,20,26)(17,24,19,25)(27,30,31,39)(28,34,35,42)(29,36,37,38)(32,40,33,41);(3,6,7,10)(4,8,5,9)(11,15,16,12)(13,19,20,17)(14,22,23,18)(21,25,26,24)(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, 8011053);