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×#58442 |
| 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 | |
4148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(4,8)(5,11)(10,15)(13,17)(19,20)(21,25)(22,29)(23,24)(26,30)(27,33)(28,31)(32,34)(35,36,39,42)(37,43,45,40)(38,41,46,48)(44,49,50,47);(6,14)(10,15)(13,17)(16,18)(20,24)(25,31)(26,29)(32,33)(35,36,39,42)(37,40,45,43)(38,41,46,48)(44,47,50,49);(3,4)(5,12)(6,10)(7,8)(9,11)(13,18)(14,15)(16,17)(19,21,23,28)(20,25,24,31)(22,27,30,34)(26,32,29,33)(35,36)(37,40)(38,41)(39,42)(43,45)(44,47)(46,48)(49,50);(3,5)(4,9)(6,13)(7,11)(8,12)(10,16)(14,17)(15,18)(19,20)(21,25)(22,26)(23,24)(27,32)(28,31)(29,30)(33,34)(35,37,39,45)(36,40,42,43)(38,44,46,50)(41,47,48,49);(3,6,7,14)(4,10,8,15)(5,13,11,17)(9,16,12,18)(19,22)(20,26)(21,27)(23,30)(24,29)(25,32)(28,34)(31,33)(35,38,39,46)(36,41,42,48)(37,44,45,50)(40,47,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011543);