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×#20708 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#117 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C2×C5 | C2×#381 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×#117 ◁ G | |
1260 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)(8,16,18,12)(9,20,22,35)(10,15)(11,14)(13,21,26,36)(17,30)(19,29,33,25)(23,27)(24,32)(28,37)(31,34);(6,8)(7,12)(9,17)(10,19)(11,18)(13,24)(14,16)(15,25)(20,34)(21,32)(22,31)(23,33)(26,28)(27,29)(30,35)(36,37);(6,9)(7,13,14,26)(8,17)(10,21)(11,22)(12,24,16,28)(15,20,27,35)(18,31)(19,32)(23,36)(25,34,29,30)(33,37);(8,18)(12,16)(17,31)(19,33)(24,28)(25,29)(30,34)(32,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13696);