Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#26852 |
| 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 | |
698 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)(8,13)(9,11)(10,12)(14,15)(16,27)(17,33)(18,37)(19,25)(20,26)(21,30)(22,35)(23,28)(24,38)(29,41)(31,40)(32,42)(34,43)(36,44)(39,45);(6,8,10,9)(7,11,12,13)(14,16,19,30,20,32,38,31)(15,21,24,27,26,40,25,42)(17,34,35,45,36,18,29,28)(22,37,33,39,41,43,44,23);(6,9,10,8)(7,11,12,13)(14,17,19,35,20,36,38,29)(15,22,24,33,26,41,25,44)(16,28,30,34,32,45,31,18)(21,39,27,43,40,23,42,37);(7,12)(11,13)(14,18,19,28,20,34,38,45)(15,23,24,37,26,39,25,43)(16,29,30,17,32,35,31,36)(21,22,27,33,40,41,42,44)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19840);