Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21484 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#120 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C4×#120 |
| prime radical | C5×#3025 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×C8 ◁ G | |
1306 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)(6,7,11,10,12,15,24,23)(8,13,19,17,18,26,33,32)(9,14,22,20,21,27,35,34)(16,28,31,37,30,25,36,29);(6,8)(7,13)(9,16)(10,17)(11,19)(12,18)(14,25)(15,26)(20,29)(21,30)(22,31)(23,32)(24,33)(27,28)(34,37)(35,36);(6,9)(7,14)(8,16)(10,20)(11,22)(12,21)(13,25)(15,27)(17,29)(18,30)(19,31)(23,34)(24,35)(26,28)(32,37)(33,36);(6,10,11,15,12,23,24,7)(8,17,19,26,18,32,33,13)(9,20,22,27,21,34,35,14)(16,29,31,28,30,37,36,25)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14472);