Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#36790 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C25 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C5×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ G | |
1548 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)(8,12)(11,13)(15,18)(19,21)(22,23,26,29)(24,30,32,27)(25,28,33,35)(31,36,37,34);(7,10)(11,13)(22,24,26,32)(23,27,29,30)(25,31,33,37)(28,34,35,36);(6,7)(8,11)(9,10)(12,13)(14,15)(16,19)(17,18)(20,21)(22,24)(23,27)(25,31)(26,32)(28,34)(29,30)(33,37)(35,36);(6,7)(8,11)(9,10)(12,13)(14,16,17,20)(15,19,18,21)(22,23)(24,27)(25,28)(26,29)(30,32)(31,34)(33,35)(36,37);(6,8)(7,11)(9,12)(10,13)(22,25,26,33)(23,28,29,35)(24,31,32,37)(27,34,30,36)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 29778);