Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#17926 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C5×#449 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C5×#431 ◁ G | |
620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,13)(9,15)(10,14)(11,16)(12,17)(18,21)(19,20)(22,23,25,29,26,28,32,35)(24,27,30,34,31,33,36,37);(6,8)(7,12)(9,17)(10,19)(11,18)(13,14)(15,21)(16,20);(6,8)(7,13)(9,19)(10,17)(11,18)(12,14)(15,20)(16,21)(22,24,25,30,26,31,32,36)(23,27,29,34,28,33,35,37);(6,9,11,10)(7,14,16,15)(8,17,18,19)(12,13,20,21)(23,28)(27,33)(29,35)(34,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10914);