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×#33744 |
| 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 | |
520 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)(17,26)(18,29)(20,31)(22,32)(23,35)(25,36)(27,43)(28,39)(30,42)(33,45)(34,40)(37,38)(41,44);(7,10)(11,13)(14,15)(16,25)(17,22)(18,23)(19,21)(20,28)(24,36)(26,32)(27,41)(29,35)(30,34)(31,39)(33,37)(38,45)(40,42)(43,44);(14,16,26,38,18,27,30,39,19,28,42,43,29,37,17,24)(15,20,32,44,23,33,34,36,21,25,40,45,35,41,22,31);(6,8,9,12)(7,11,10,13)(15,21)(20,25)(22,34)(23,35)(31,36)(32,40)(33,41)(44,45)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26732);