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×#17919 |
| 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×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C5×#431 ◁ G | |
596 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)(7,11)(9,16)(12,18)(13,17)(15,20)(19,21)(22,23,25,29,26,28,32,35)(24,27,30,34,31,33,36,37);(6,7,10,13)(8,12,14,19)(9,17,16,11)(15,21,20,18);(6,8)(7,12)(9,15)(10,14)(11,18)(13,19)(16,20)(17,21)(22,24,25,30,26,31,32,36)(23,27,29,34,28,33,35,37);(8,14)(12,19)(15,20)(18,21)(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, 10907);