Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#2218 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
508 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)(8,11)(9,10)(12,13)(16,24)(17,28)(18,29)(20,31)(22,37)(23,36)(25,30)(26,41)(27,40)(32,35)(33,42)(34,45)(38,39)(43,44);(7,10)(11,13)(14,15)(16,20)(17,22)(18,23)(19,21)(24,31)(25,32)(26,34)(27,33)(28,37)(29,36)(30,35)(38,43)(39,44)(40,42)(41,45);(14,16,25,41,18,26,17,24,19,27,28,39,29,38,30,40)(15,20,32,45,23,34,22,31,21,33,37,44,36,43,35,42);(6,8,9,12)(7,11,10,13)(15,21)(20,33)(22,35)(23,36)(31,42)(32,37)(34,43)(44,45)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26730);