Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#36698 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
1484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15)(16,18)(17,19)(20,21)(22,23,26,27)(24,28,31,34)(25,29,33,35)(30,36,32,37);(14,16,17,21)(15,18,19,20)(23,27)(28,34)(29,35)(36,37);(7,10)(11,13)(14,15)(16,20)(17,19)(18,21)(22,24,26,31)(23,28,27,34)(25,32,33,30)(29,37,35,36);(15,19)(18,20)(22,25,26,33)(23,29,27,35)(24,30,31,32)(28,36,34,37);(6,8,9,12)(7,11,10,13)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 29686);