Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#20671 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#117 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#380 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C5×M16 ◁ G | |
1324 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)(8,16)(9,20,22,13)(10,23)(11,15)(12,19)(14,24)(17,30,31,28)(18,29)(21,27,36,35)(25,32,34,37)(26,33);(6,8,10,19,11,18,24,33)(7,12,14,16,15,26,23,29)(9,17,21,32,22,31,36,37)(13,25,27,28,20,34,35,30);(6,9)(7,13)(8,17)(10,21)(11,22)(12,25)(14,27)(15,20)(16,28)(18,31)(19,32)(23,35)(24,36)(26,34)(29,30)(33,37);(8,18)(12,26)(16,29)(17,31)(19,33)(25,34)(28,30)(32,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13659);