Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C4 |
| direct product | C5×#12014 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C2×D4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×C4⋊C8 |
| prime radical | C23×C5 | C4⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ C2×C4×C5×C8 ◁ G | |
220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7)(8,15)(9,13)(10,19)(11,14)(12,17)(16,20)(18,21)(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);(6,8,10,17,11,18,9,16)(7,12,13,15,14,20,19,21)(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);(6,9,11,10)(7,13,14,19)(8,16,18,17)(12,15,20,21)(24,32)(28,38)(33,44)(34,45)(39,48)(40,49)(46,52)(50,53)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5002);