Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33878 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3280 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×D4 ◁ C5×C8×D4 ◁ G | |
562 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)(6,7,9,11)(8,12,13,10)(14,15)(16,21)(17,27)(18,31)(19,35)(20,24)(22,29)(23,33)(25,41)(26,40)(28,39)(30,43)(32,36)(34,45)(37,44)(38,42);(6,8)(7,10)(9,13)(11,12)(14,16,20,19)(15,21,24,35)(17,28,30,25)(18,26,34,33)(22,37,38,36)(23,31,40,45)(27,39,43,41)(29,44,42,32);(14,17)(15,22)(16,25)(18,32)(19,28)(20,30)(21,36)(23,39)(24,38)(26,42)(27,31)(29,33)(34,44)(35,37)(40,41)(43,45);(8,13)(10,12)(14,16,20,19)(15,21,24,35)(17,25,30,28)(18,26,34,33)(22,36,38,37)(23,31,40,45)(27,41,43,39)(29,32,42,44)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26866);