Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21478 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#120 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C4×#120 |
| prime radical | C5×#3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×C8 ◁ G | |
1282 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)(6,7,12,17,13,18,11,24)(8,19,22,14,23,33,28,32)(9,15,25,10,16,26,29,27)(20,21,35,37,31,34,36,30);(6,8)(7,14)(9,20)(10,21)(11,28)(12,22)(13,23)(15,30)(16,31)(17,33)(18,32)(19,24)(25,35)(26,37)(27,34)(29,36);(6,9)(7,15)(8,20)(10,17)(11,29)(12,25)(13,16)(14,30)(18,26)(19,34)(21,33)(22,35)(23,31)(24,27)(28,36)(32,37);(6,10,12,26,13,27,11,15)(7,16,17,29,18,9,24,25)(8,21,22,37,23,34,28,30)(14,31,33,36,32,20,19,35)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14466);