Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60338 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C25 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C24×C4 |
| prime radical | C5×#3531 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 53 points:
(1,2,3,4,5)(6,7)(8,14)(9,12)(10,13)(11,16)(15,20)(17,19)(18,21)(22,23,26,31,27,29,38,42)(24,28,34,41,32,39,46,50)(25,35,36,49,37,30,43,44)(33,47,48,53,45,40,51,52);(6,8,10,16)(7,11,13,14)(9,17,18,15)(12,20,21,19)(22,24)(23,28)(25,33)(26,34)(27,32)(29,39)(30,40)(31,41)(35,47)(36,48)(37,45)(38,46)(42,50)(43,51)(44,52)(49,53);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(23,29)(24,32)(30,35)(31,42)(33,45)(34,46)(44,49)(48,51);(7,13)(11,14)(12,21)(19,20)(24,32)(28,39)(33,45)(34,46)(40,47)(41,50)(48,51)(52,53);(22,25,26,36,27,37,38,43)(23,30,31,44,29,35,42,49)(24,33,34,48,32,45,46,51)(28,40,41,52,39,47,50,53)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53326);