Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21477 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #120 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C4×#120 |
| prime radical | C5×#3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,13,16,12,17,11,27)(8,18,21,14,22,31,28,32)(9,23,24,26,25,15,29,10)(19,33,35,30,36,20,34,37);(6,8)(7,14)(9,19)(10,20)(11,28)(12,22)(13,21)(15,30)(16,31)(17,32)(18,27)(23,37)(24,35)(25,36)(26,33)(29,34);(6,9)(7,15)(8,19)(10,16)(11,29)(12,25)(13,24)(14,30)(17,23)(18,33)(20,31)(21,35)(22,36)(26,27)(28,34)(32,37);(6,10,13,23,12,26,11,15)(7,9,16,24,17,25,27,29)(8,20,21,37,22,33,28,30)(14,19,31,35,32,36,18,34)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14465);