Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19051 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3545 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
950 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)(7,10)(8,12)(11,13)(14,15,20,25)(16,26,29,38)(17,30,31,39)(18,24,19,23)(21,33,36,28)(22,34,37,32)(27,35,41,45)(40,42,44,43);(8,12)(10,13)(14,16)(15,21)(17,27)(18,33)(19,28)(20,29)(22,35)(23,38)(24,26)(25,36)(30,44)(31,41)(32,43)(34,42)(37,45)(39,40);(6,7)(8,13)(9,11)(10,12)(14,17)(15,22)(16,27)(18,32)(19,34)(20,31)(21,35)(23,30)(24,39)(25,37)(26,40)(28,42)(29,41)(33,43)(36,45)(38,44);(17,31)(22,37)(27,41)(30,39)(32,34)(35,45)(40,44)(42,43)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12039);