Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#20612 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#120 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#379 |
| prime radical | C2×C5×#585 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
3640 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)(6,7,10,15,11,14,23,28)(8,16,19,29,18,12,22,25)(9,13,21,27,20,26,33,36)(17,24,32,35,31,30,34,37);(6,8)(7,12)(9,17)(10,22)(11,18)(13,24)(14,16)(15,29)(19,23)(20,31)(21,34)(25,28)(26,30)(27,37)(32,33)(35,36);(6,9)(7,13)(8,17)(10,21)(11,20)(12,24)(14,26)(15,27)(16,30)(18,31)(19,32)(22,34)(23,33)(25,35)(28,36)(29,37);(7,14)(12,16)(13,26)(15,28)(24,30)(25,29)(27,36)(35,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13600);