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×#60296 |
| 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)(22,23,26,30,27,31,37,43)(24,28,34,40,33,39,46,50)(25,29,36,42,35,41,47,51)(32,38,45,49,44,48,52,53);(6,8)(7,11)(9,16)(10,15)(12,14)(13,18)(17,21)(19,20)(22,24)(23,28)(25,32)(26,34)(27,33)(29,38)(30,40)(31,39)(35,44)(36,45)(37,46)(41,48)(42,49)(43,50)(47,52)(51,53);(8,15)(11,14)(16,21)(18,20)(22,25)(23,29)(24,32)(26,36)(27,35)(28,38)(30,42)(31,41)(33,44)(34,45)(37,47)(39,48)(40,49)(43,51)(46,52)(50,53);(7,12)(11,14)(13,19)(18,20)(25,35)(29,41)(32,44)(36,47)(38,48)(42,51)(45,52)(49,53);(6,9,10,17)(7,13,12,19)(8,16,15,21)(11,18,14,20)(24,33)(28,39)(32,44)(34,46)(38,48)(40,50)(45,52)(49,53)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53284);