Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#12895 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | C23 |
| commutator | C23 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle, prime radical | C23×C5 | C22×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ G | |
298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7)(8,13)(9,11)(10,12)(14,15)(16,18)(17,19)(20,21)(22,23,25,29,26,30,35,41,27,31,36,42,37,43,47,51)(24,28,33,39,34,40,46,50,32,38,44,48,45,49,52,53);(6,8,10,9)(7,11,12,13)(14,16,17,20)(15,18,19,21)(22,24)(23,28)(25,33)(26,34)(27,32)(29,39)(30,40)(31,38)(35,46)(36,44)(37,45)(41,50)(42,48)(43,49)(47,52)(51,53);(6,9,10,8)(7,11,12,13)(15,19)(18,21)(24,32)(28,38)(33,44)(34,45)(39,48)(40,49)(46,52)(50,53)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5883);