Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#60332 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| 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×C5×C8 ◁ 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)(22,23,26,31,27,29,37,42)(24,28,34,40,32,38,45,49)(25,30,35,43,36,41,47,51)(33,39,46,50,44,48,52,53);(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)(29,38)(30,39)(31,40)(35,46)(36,44)(37,45)(41,48)(42,49)(43,50)(47,52)(51,53);(6,9,10,18)(7,12,13,21)(8,15,16,17)(11,19,14,20)(23,29)(24,32)(30,41)(31,42)(33,44)(34,45)(43,51)(46,52);(7,13)(11,14)(12,21)(19,20)(24,32)(28,38)(33,44)(34,45)(39,48)(40,49)(46,52)(50,53);(22,25,26,35,27,36,37,47)(23,30,31,43,29,41,42,51)(24,33,34,46,32,44,45,52)(28,39,40,50,38,48,49,53)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53320);