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×#22259 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C23×C5 | C2×#41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G | |
594 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)(8,15)(9,13)(10,19)(11,14)(12,17)(16,21)(18,20)(22,23,27,31,28,32,26,36)(24,29,34,37,33,30,25,35);(6,8)(7,12)(9,18)(10,16)(11,17)(13,21)(14,15)(19,20)(22,24,27,34,28,33,26,25)(23,29,31,37,32,30,36,35);(6,9,11,10)(7,13,14,19)(8,16,17,18)(12,20,15,21)(22,25,26,33,28,34,27,24)(23,30,31,35,32,29,36,37);(6,10,11,9)(7,13,14,19)(8,16,17,18)(12,21,15,20)(24,33)(25,34)(29,30)(35,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15247);