Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18655 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C2×C5×#41 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
884 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)(8,16,19,20)(9,13,10,14)(12,18,21,17)(23,28)(24,32)(25,26)(30,31)(33,34)(35,36);(6,8)(7,12)(9,17)(10,18)(11,19)(13,20)(14,16)(15,21)(22,23,27,31)(24,29,33,37)(25,28,26,30)(32,35,34,36);(6,9,11,10)(7,13,15,14)(8,17,19,18)(12,20,21,16)(22,24,27,33)(23,29,31,37)(25,32,26,34)(28,35,30,36);(6,9,11,10)(7,14,15,13)(8,18,19,17)(12,20,21,16)(22,25,27,26)(23,30,31,28)(24,32,33,34)(29,36,37,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11643);