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×#60297 |
| 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×#3455 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ G | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 53 points:
(1,2,3,4,5)(6,7)(8,14)(9,13)(10,12)(11,15)(16,20)(17,19)(18,21)(22,23,26,30,27,31,38,45)(24,28,34,41,33,40,47,51)(25,29,37,44,36,43,49,53)(32,39,46,50,35,42,48,52);(6,8)(7,11)(9,16)(10,15)(12,14)(13,18)(17,21)(19,20)(22,24)(23,28)(25,35)(26,34)(27,33)(29,42)(30,41)(31,40)(32,36)(37,48)(38,47)(39,43)(44,52)(45,51)(46,49)(50,53);(8,15)(11,14)(16,21)(18,20)(22,25)(23,29)(24,32)(26,37)(27,36)(28,39)(30,44)(31,43)(33,35)(34,46)(38,49)(40,42)(41,50)(45,53)(47,48)(51,52);(7,12)(11,14)(13,19)(18,20)(25,36)(29,43)(32,35)(37,49)(39,42)(44,53)(46,48)(50,52);(6,9,10,17)(7,13,12,19)(8,16,15,21)(11,18,14,20)(24,33)(28,40)(32,35)(34,47)(39,42)(41,51)(46,48)(50,52)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53285);