Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#17990 |
| 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×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C5×#431 ◁ G | |
584 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)(6,7)(8,14)(9,17)(10,12)(11,15)(13,19)(16,21)(18,20)(22,23,26,30,27,29,35,36)(24,28,32,25,31,33,37,34);(6,8,10,16)(7,11,12,20)(9,18,19,15)(13,21,17,14)(22,24)(23,28)(25,30)(26,32)(27,31)(29,33)(34,36)(35,37);(6,7)(8,11)(9,13)(10,12)(14,15)(16,20)(17,19)(18,21)(22,25,26,33,27,34,35,28)(23,24,30,32,29,31,36,37);(7,12)(11,20)(13,17)(14,21)(23,29)(24,31)(30,36)(32,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10978);