Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26403 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 45 points:
(1,2,3,4,5)(6,7,10,12)(8,11,9,13)(14,15)(16,27)(17,22)(18,35)(19,25)(20,26)(21,30)(23,34)(24,37)(28,41)(29,38)(31,39)(32,45)(33,42)(36,40)(43,44);(6,8,10,9)(7,11,12,13)(14,16)(15,21)(17,32)(18,29)(19,36)(20,31)(22,41)(23,38)(24,39)(25,27)(26,40)(28,34)(30,37)(33,44)(35,43)(42,45);(6,9,10,8)(7,11,12,13)(14,17,20,18)(15,22,26,35)(16,28,31,44)(19,33,37,34)(21,38,40,45)(23,25,42,24)(27,41,39,43)(29,36,32,30);(14,18,20,17)(15,23,26,42)(16,29,31,32)(19,33,37,34)(21,38,40,45)(22,25,35,24)(27,43,39,41)(28,36,44,30)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19391);