Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13469 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#13469 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2065 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,14,18,13,19,12,32)(8,20,9,24,23,33,28,35)(10,29,31,37,11,30,25,17)(15,27,36,21,34,26,16,22);(6,8,10,21,12,9,25,26,13,23,11,22,14,28,31,27)(7,15,17,20,18,16,29,35,19,34,37,33,32,36,30,24);(6,9)(7,16)(8,12)(10,22)(11,21)(13,28)(14,23)(15,18)(17,33)(19,36)(20,37)(24,30)(25,26)(27,31)(29,35)(32,34)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6457);