Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7694 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#7729 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C43×C5 ◁ G | |
346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 53 points:
(1,2,3,4,5)(6,7)(8,14,16,11)(9,17)(10,13)(12,18)(15,19,21,20)(22,23)(24,30,32,37)(25,29)(26,28)(27,31,35,33)(34,36)(38,39,41,45,42,44,50,52)(40,46,48,53,47,43,49,51);(6,8)(7,11)(9,15)(10,16)(12,19)(13,14)(17,20)(18,21)(22,24,26,32)(23,27,28,35)(25,33,34,31)(29,37,36,30)(38,40)(39,43)(41,49)(42,47)(44,46)(45,53)(48,50)(51,52);(6,9,10,18)(7,12,13,17)(8,15,16,21)(11,19,14,20)(23,28)(27,35)(29,36)(30,37)(39,44)(40,47)(45,52)(48,49)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 717);