Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33605 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33750 |
| 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 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
516 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)(14,15)(16,21)(17,27)(18,32)(19,34)(20,24)(22,31)(23,29)(25,39)(26,38)(28,41)(30,42)(33,45)(35,40)(36,43)(37,44);(7,10)(11,13)(14,16,20,19)(15,21,24,34)(17,28,30,25)(18,26,33,29)(22,36,37,35)(23,32,38,45)(27,41,42,39)(31,43,44,40);(14,17,29,43,19,28,33,44,20,30,26,40,16,25,18,31)(15,22,32,39,21,35,38,42,24,37,45,41,34,36,23,27);(6,8,9,12)(7,11,10,13)(17,30)(22,37)(25,28)(27,42)(31,44)(35,36)(39,41)(40,43)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26738);