Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18471 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3560 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,16)(8,13,18,21)(9,15,10,14)(12,19,20,17)(23,27)(25,32)(28,36)(29,33)(31,34)(35,37);(6,8)(7,12)(9,17)(10,19)(11,18)(13,14)(15,21)(16,20)(22,23)(24,28)(25,33)(26,29)(27,32)(30,37)(31,35)(34,36);(6,8)(7,13)(9,19)(10,17)(11,18)(12,14)(15,20)(16,21)(22,24)(23,28)(25,34)(26,30)(27,35)(29,37)(31,32)(33,36);(6,9,11,10)(7,14,16,15)(8,17,18,19)(12,13,20,21)(24,30)(28,37)(31,34)(35,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11459);