Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13529 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
342 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5)(10,11,13,15)(12,16,17,14)(18,19,21,26,35,43,46,39,23,28,36,38,22,27,33,24)(20,29,30,44,37,49,48,41,32,45,47,40,31,42,34,25);(6,7)(8,9)(10,12,13,17)(11,14,15,16)(19,24)(21,33)(22,35)(25,29)(26,27)(28,39)(30,34)(31,37)(36,46)(38,43)(40,49)(41,45)(42,44)(47,48);(7,9)(18,20,23,32)(19,25,28,41)(21,34,36,48)(22,37,35,31)(24,29,39,45)(26,42,38,49)(27,44,43,40)(30,46,47,33)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6517);