Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29602 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#29674 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 49 points:
(1,2,3,4,5)(10,11)(12,15)(13,14)(16,17)(18,19,23,27)(20,24,29,38)(21,31,33,41)(22,34,36,26)(25,35,40,32)(28,42,43,49)(30,46,45,39)(37,47,48,44);(6,7)(8,9)(11,14)(15,17)(18,20,23,29)(19,24,27,38)(21,28,33,43)(22,30,36,45)(25,37,40,48)(26,39,34,46)(31,42,41,49)(32,44,35,47);(7,9)(12,16)(15,17)(18,21,23,33)(19,25,27,40)(20,28,29,43)(22,35,36,32)(24,37,38,48)(26,41,34,31)(30,47,45,44)(39,49,46,42);(10,12,13,16)(11,15,14,17)(20,29)(24,38)(28,43)(30,45)(37,48)(39,46)(42,49)(44,47)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22662);