Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#24461 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
1318 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)(12,21)(13,27)(14,30)(15,19)(16,20)(17,24)(18,28)(22,26)(23,32)(25,33)(29,34)(31,35)(36,38)(37,40)(39,41);(10,12,16,26)(11,17,20,33)(13,23,29,39)(14,25,15,24)(18,32,35,41)(19,22,30,21)(27,36,34,37)(28,40,31,38);(7,9)(10,11)(12,22)(13,18)(14,19)(15,30)(16,20)(17,24)(21,26)(23,37)(25,33)(27,28)(29,35)(31,34)(32,40)(36,39)(38,41);(10,13,16,29)(11,18,20,35)(12,23,26,39)(14,31,15,28)(17,32,33,41)(19,34,30,27)(21,36,22,37)(24,40,25,38)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 17449);