Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#11641 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C2×C22⋊C8 |
| prime radical | C23×C5 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
292 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)(22,23,26,31,27,32,41,49,28,33,42,50,25,30,39,47)(24,34,36,51,35,45,52,48,37,46,40,53,38,29,43,44);(6,8,11,18)(7,12,14,20)(9,17,10,16)(13,21,19,15)(22,24)(23,29)(25,38)(26,40)(27,35)(28,37)(30,46)(31,48)(32,34)(33,45)(36,42)(39,52)(41,43)(44,50)(47,51)(49,53);(6,9,11,10)(7,13,14,19)(8,16,18,17)(12,15,20,21)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4629);