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×#26855 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#2002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
764 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,13)(9,11)(10,12)(14,15)(16,17)(18,24)(19,23)(20,25)(21,22)(26,28)(27,29);(6,8,10,9)(7,11,12,13)(14,15,19,24,20,25,18,23)(16,17,22,28,27,29,26,21);(6,9,10,8)(7,11,12,13)(14,16,18,26,20,27,19,22)(15,21,23,29,25,28,24,17);(7,12)(11,13)(14,17,18,21,20,29,19,28)(15,22,23,16,25,26,24,27)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19843);