Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#23520 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| prime radical | C22×C5×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
1350 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)(13,25)(14,19)(15,30)(16,20)(17,23)(18,29)(22,38)(24,33)(26,37)(27,40)(28,36)(31,34)(32,41)(35,39);(10,12)(11,17)(13,26)(14,23)(15,31)(16,24)(18,35)(19,33)(20,34)(21,30)(22,29)(25,32)(27,41)(28,39)(36,37)(38,40);(7,9)(10,13,15,28,16,29,14,27)(11,18,19,36,20,25,30,40)(12,22,23,39,24,26,31,41)(17,32,21,37,34,35,33,38);(10,14,16,15)(11,19,20,30)(12,23,24,31)(13,27,29,28)(17,33,34,21)(18,36,25,40)(22,39,26,41)(32,38,35,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16508);