Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23646 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#121 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C4×C5 ◁ G | |
906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,11)(9,10)(12,13)(14,15,18,23,20,25,19,24)(16,21,27,36,28,37,34,41)(17,29,32,40,33,39,31,22)(26,38,30,44,43,45,42,35);(7,10)(11,13)(14,16)(15,21)(17,30)(18,27)(19,34)(20,28)(22,38)(23,36)(24,41)(25,37)(26,31)(29,44)(32,43)(33,42)(35,39)(40,45);(6,8,9,12)(7,11,10,13)(14,17)(15,22)(16,26)(18,31)(19,32)(20,33)(21,35)(23,39)(24,29)(25,40)(27,42)(28,43)(30,34)(36,45)(37,44)(38,41);(14,18,20,19)(15,23,25,24)(16,27,28,34)(17,31,33,32)(21,36,37,41)(22,39,40,29)(26,42,43,30)(35,45,44,38)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16634);