Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#8122 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#95 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C24×C5 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C42×C5 ◁ G | |
520 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,15,17,12)(9,19)(10,13)(11,14)(16,20,18,21)(22,23,26,30,27,31,25,29)(24,32,35,36,34,28,33,37);(6,8,10,18,11,17,9,16)(7,12,13,20,14,15,19,21)(22,24)(23,28,31,32)(25,33)(26,35)(27,34)(29,36,30,37);(6,9,11,10)(7,13,14,19)(8,16,17,18)(12,20,15,21)(22,25,27,26)(23,29,31,30)(24,33,34,35)(28,36,32,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1110);