Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C5×#23425 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C5×#117 ◁ G | |
1938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7)(8,9)(14,15,20,25)(16,26,28,37)(17,31,34,38)(18,24,19,23)(21,30,36,29)(22,32,39,33)(27,41,42,35)(40,43,45,44);(7,9)(10,11)(12,13)(14,16)(15,21)(17,27)(18,30)(19,29)(20,28)(22,35)(23,37)(24,26)(25,36)(31,45)(32,44)(33,43)(34,42)(38,40)(39,41);(11,13)(14,17,20,34)(15,22,25,39)(16,27,28,42)(18,33,19,32)(21,35,36,41)(23,31,24,38)(26,40,37,45)(29,44,30,43);(16,28)(21,36)(26,37)(27,42)(29,30)(35,41)(40,45)(43,44)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16413);