Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#12592 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | C23 |
| commutator | C22 | C2×C4×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle, prime radical | C23×C5 | C22×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ G | |
358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 49 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 49 points:
(1,2,3,4,5)(10,11)(12,15)(13,14)(16,17)(18,19,21,25,22,26,31,37,23,27,32,38,33,39,43,47)(20,24,29,35,30,36,42,46,28,34,40,44,41,45,48,49);(6,7,8,9)(11,14)(15,17)(18,20)(19,24)(21,29)(22,30)(23,28)(25,35)(26,36)(27,34)(31,42)(32,40)(33,41)(37,46)(38,44)(39,45)(43,48)(47,49);(10,12,13,16)(11,15,14,17)(20,28)(24,34)(29,40)(30,41)(35,44)(36,45)(42,48)(46,49)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5580);