Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#22370 |
| 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 | C22×C5×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
690 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)(7,10)(8,12)(14,15)(16,20)(17,19)(18,21)(22,23,27,24,26,29,35,31)(25,32,34,37,33,28,36,30);(6,7)(8,11)(9,10)(12,13)(14,16)(15,18)(17,21)(19,20)(22,24,27,29,26,31,35,23)(25,30,34,32,33,37,36,28);(7,10)(11,13)(15,19)(16,21)(22,25)(23,28)(24,30)(26,33)(27,34)(29,32)(31,37)(35,36);(6,8,9,12)(7,11,10,13)(15,19)(18,20)(25,33)(28,32)(30,37)(34,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15358);