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×#22304 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle, prime radical | C23×C5 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(7,10)(8,12)(14,15)(16,19)(17,21)(18,20)(22,23,28,24,27,31,26,30)(25,29,33,32,34,36,35,37);(6,7)(8,11)(9,10)(12,13)(14,16,18,17)(15,19,20,21)(22,24,28,31,27,30,26,23)(25,32,33,36,34,37,35,29);(7,10)(11,13)(14,17,18,16)(15,19,20,21)(22,25)(23,29)(24,32)(26,35)(27,34)(28,33)(30,37)(31,36);(6,8,9,12)(7,11,10,13)(22,26,27,28)(23,30,31,24)(25,33,34,35)(29,32,36,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 15292);