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×#23522 |
| 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)(12,20)(13,24)(14,28)(15,18)(16,22)(17,27)(19,30)(21,37)(23,35)(25,36)(26,34)(29,32)(31,40)(33,38)(39,41);(10,12,15,23)(11,16,18,32)(13,25,27,21)(14,29,30,22)(17,33,24,31)(19,35,28,20)(26,38,41,40)(34,36,39,37);(7,9)(10,13,14,26,15,27,30,41)(11,17,19,34,18,24,28,39)(12,21,22,38,23,25,29,40)(16,31,20,36,32,33,35,37);(11,18)(16,32)(17,24)(19,28)(20,35)(31,33)(34,39)(36,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16510);