Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#33787 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C5×#117 ◁ C5×#405 ◁ G | |
586 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)(8,12)(9,11)(10,13)(16,24)(17,27)(18,29)(20,32)(22,37)(23,36)(25,42)(26,41)(28,31)(30,40)(33,44)(34,45)(35,43)(38,39);(6,8,9,13)(7,10,11,12)(14,15)(16,20)(17,22)(18,23)(19,21)(24,32)(25,34)(26,33)(27,37)(28,38)(29,36)(30,35)(31,39)(40,43)(41,44)(42,45);(8,13)(10,12)(14,16,19,26)(15,20,21,33)(17,28,30,42)(18,31,29,25)(22,38,35,45)(23,39,36,34)(24,40,41,27)(32,43,44,37);(7,11)(10,12)(15,21)(20,33)(22,35)(23,36)(32,44)(34,39)(37,43)(38,45)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26775);