Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#17607 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3494 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C5×#117 ◁ G | |
990 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)(8,14,16,11)(9,12)(10,13)(15,20,21,18)(17,19)(22,23)(24,29)(25,30)(26,35)(27,31)(28,32)(33,36)(34,37);(6,8)(7,11)(9,15)(10,16)(12,18)(13,14)(17,21)(19,20)(22,24)(23,28)(25,32)(26,34)(27,33)(29,35)(30,36)(31,37);(6,9,10,17)(7,12,13,19)(8,15,16,21)(11,18,14,20)(22,24)(23,29)(25,34)(26,32)(27,33)(28,35)(30,37)(31,36);(22,25,27,26)(23,30,31,35)(24,32,33,34)(28,36,37,29)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10595);