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×#58488 |
| 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 | |
4186 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);(3,4,7,10)(5,11,13,8)(6,9,14,16)(12,17,18,15)(19,20)(21,27)(22,29)(23,31)(24,25)(26,32)(28,30)(33,34)(35,36)(37,40)(38,46)(39,42)(41,47)(43,45)(44,49)(48,50);(6,14)(9,16)(12,18)(15,17)(20,25)(22,27)(23,32)(28,34)(35,36,39,42)(37,43,45,40)(38,46,47,41)(44,48,50,49);(3,5,7,13)(4,8,10,11)(6,12,14,18)(9,15,16,17)(19,21,24,29)(20,22,25,27)(23,28,32,34)(26,30,31,33)(35,36)(37,43)(38,41)(39,42)(40,45)(44,49)(46,47)(48,50);(3,4)(5,11)(6,9)(7,10)(8,13)(12,17)(14,16)(15,18)(19,22)(20,21)(23,30)(24,27)(25,29)(26,28)(31,34)(32,33)(35,37,39,45)(36,40,42,43)(38,44,47,50)(41,48,46,49);(3,6,7,14)(4,9,10,16)(5,12,13,18)(8,15,11,17)(19,23)(20,26)(21,28)(22,30)(24,32)(25,31)(27,33)(29,34)(35,38,39,47)(36,41,42,46)(37,44,45,50)(40,48,43,49)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8011459);