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×#36700 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| socle | C23×C5 | C25 |
| prime radical | C24×C5 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C5 ◁ G | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,13)(10,12)(14,15)(16,20)(17,21)(18,19)(22,25)(23,27)(24,26)(28,29)(30,31,34,37)(32,35,39,43)(33,36,41,45)(38,42,40,44);(6,8,10,9)(7,11,12,13)(15,19)(20,25)(21,26)(27,29)(30,32,34,39)(31,35,37,43)(33,40,41,38)(36,44,45,42);(7,12)(11,13)(14,16)(15,20)(17,23)(18,22)(19,25)(21,27)(24,28)(26,29)(30,33)(31,36)(32,38)(34,41)(35,42)(37,45)(39,40)(43,44);(6,9,10,8)(7,13,12,11)(16,22)(20,25)(23,28)(27,29)(31,37)(33,41)(35,43)(38,40);(14,17,18,24)(15,21,19,26)(16,23,22,28)(20,27,25,29)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 29688);