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×#18657 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3511 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
940 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)(22,23)(24,32)(25,28)(26,30)(27,31)(29,33)(34,37)(35,36);(6,8,11,19)(7,12,15,21)(9,17,10,18)(13,20,14,16)(23,28)(25,26)(29,36)(30,31)(32,37)(33,34);(6,9,11,10)(7,13,15,14)(8,17,19,18)(12,20,21,16)(22,24,27,35)(23,29,31,37)(25,33,26,34)(28,36,30,32);(6,9,11,10)(7,14,15,13)(8,18,19,17)(12,20,21,16)(22,25,27,26)(23,30,31,28)(24,33,35,34)(29,32,37,36)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11645);