Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18719 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
928 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)(7,10)(8,12)(11,13)(14,15,20,25)(16,26,29,39)(17,22,31,37)(18,24,19,23)(21,33,35,28)(27,40,43,44)(30,38,32,36)(34,41,45,42);(8,12)(10,13)(14,16)(15,21)(17,27)(18,28)(19,33)(20,29)(22,34)(23,26)(24,39)(25,35)(30,41)(31,43)(32,42)(36,44)(37,45)(38,40);(6,7)(8,13)(9,11)(10,12)(14,17,20,31)(15,22,25,37)(16,27,29,43)(18,32,19,30)(21,34,35,45)(23,38,24,36)(26,40,39,44)(28,42,33,41);(14,18,20,19)(15,23,25,24)(16,28,29,33)(17,30,31,32)(21,26,35,39)(22,36,37,38)(27,41,43,42)(34,44,45,40)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11707);