Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #20338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#20338 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22×C42 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C22×#95 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ G | |
2036 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)(8,16,19,29)(9,13)(10,14)(11,15)(12,18,24,32)(17,28,30,36)(20,25)(21,26)(22,27)(23,31,34,37)(33,35);(6,8)(7,12,15,24)(9,17)(10,18)(11,19)(13,23,25,34)(14,16,27,29)(20,30)(21,31)(22,32)(26,28,35,36)(33,37);(6,9)(7,13)(8,17)(10,21)(11,20)(12,23)(14,26)(15,25)(16,28)(18,31)(19,30)(22,33)(24,34)(27,35)(29,36)(32,37);(9,20)(13,25)(17,30)(21,33)(23,34)(26,35)(28,36)(31,37)410 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13326);