Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#17986 |
| 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×#121 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
578 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)(6,7,11,15)(8,16,18,21)(9,14,10,13)(12,19,20,17)(22,23,26,30,27,29,35,36)(24,28,32,25,31,33,37,34);(6,8)(7,12)(9,17)(10,19)(11,18)(13,16)(14,21)(15,20)(22,24)(23,28)(25,30)(26,32)(27,31)(29,33)(34,36)(35,37);(22,25,26,33,27,34,35,28)(23,24,30,32,29,31,36,37);(6,9,11,10)(7,13,15,14)(8,17,18,19)(12,16,20,21)(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, 10974);