Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23532 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 41 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 41 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,22)(13,27)(14,20)(15,19)(16,21)(17,24)(18,31)(23,32)(25,34)(26,33)(28,40)(29,36)(30,35)(37,39)(38,41);(10,12,16,26)(11,17,21,34)(13,28,29,23)(14,25,15,24)(18,32,35,40)(19,33,20,22)(27,37,36,41)(30,38,31,39);(7,9)(10,13,14,30,16,29,15,31)(11,18,19,36,21,35,20,27)(12,23,24,38,26,28,25,39)(17,32,22,37,34,40,33,41);(13,29)(18,35)(23,28)(27,36)(30,31)(32,40)(37,41)(38,39)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16520);