Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#22290 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C23×C4 |
| prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
702 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)(8,12)(9,11)(10,13)(14,15)(16,19)(17,21)(18,20)(22,23,27,24,26,29,35,31)(25,28,34,30,33,36,37,32);(6,8)(7,10)(9,13)(11,12)(14,16,18,17)(15,19,20,21)(22,24,27,29,26,31,35,23)(25,32,34,28,33,30,37,36);(8,13)(10,12)(14,17,18,16)(15,19,20,21)(22,25)(23,28)(24,30)(26,33)(27,34)(29,36)(31,32)(35,37);(7,11)(10,12)(23,29)(24,31)(25,33)(34,37)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15278);