Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17624 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×Q8 ◁ G | |
898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,16)(9,14)(10,13)(11,15)(12,17)(18,20)(19,21)(23,27,29,32)(28,34,35,37);(6,8)(7,12)(9,17)(10,19)(11,18)(13,16)(14,20)(15,21)(22,23)(24,28)(25,32)(26,29)(27,33)(30,35)(31,37)(34,36);(6,9,11,10)(7,13,15,14)(8,17,18,19)(12,16,21,20)(22,24)(23,28)(25,31)(26,30)(27,34)(29,35)(32,37)(33,36);(6,9,11,10)(7,14,15,13)(8,17,18,19)(12,20,21,16)(24,30)(28,35)(31,36)(34,37)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10612);