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×#58477 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 50 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 50 points:
(1,2);(4,8)(5,11)(10,15)(12,17)(19,20)(21,24)(22,27)(23,28)(25,29)(26,31)(30,32)(33,34)(35,36,41,44)(37,42,38,39)(40,48,49,43)(45,47,46,50);(4,8)(9,14)(10,15)(16,18)(20,21)(22,29)(26,28)(30,34)(35,37,41,38)(36,39,44,42)(40,46,49,45)(43,50,48,47);(3,4)(5,9)(6,10)(7,8)(11,14)(12,16)(13,15)(17,18)(19,21,24,20)(22,27,29,25)(23,28,31,26)(30,33,34,32)(35,38,41,37)(36,42,44,39)(40,46,49,45)(43,50,48,47);(3,5,7,11)(4,9,8,14)(6,12,13,17)(10,16,15,18)(19,22)(20,25)(21,27)(23,30)(24,29)(26,32)(28,33)(31,34)(35,39,41,42)(36,38,44,37)(40,47,49,50)(43,46,48,45);(3,6,7,13)(4,10,8,15)(5,12,11,17)(9,16,14,18)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)(35,40)(36,43)(37,45)(38,46)(39,47)(41,49)(42,50)(44,48)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011385);