Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12506 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C8 |
| direct product | C5×#12514 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23×C5 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ C5×C82 ◁ G | |
246 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)(9,17)(10,12)(11,15)(13,18)(16,20)(19,21)(22,23,30,37)(24,38,43,32)(25,44,39,31)(26,34,45,53)(27,33,47,51)(28,36,29,35)(40,50,46,49)(41,52,42,48);(6,8,9,15,10,16,18,21)(7,11,13,14,12,19,17,20)(22,24,27,40,28,41,45,25,30,43,47,46,29,42,26,39)(23,31,34,48,35,49,51,32,37,44,53,52,36,50,33,38);(7,12)(11,19)(13,17)(14,20)(22,25,45,42,29,40,27,43,30,39,26,41,28,46,47,24)(23,32,51,50,36,48,34,44,37,38,33,49,35,52,53,31)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5502);