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×#26996 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ G | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15)(16,23)(17,28)(18,21)(19,22)(20,25)(24,26)(27,29);(6,8,9,13)(7,10,11,12)(14,15,18,28,19,22,17,21)(16,20,26,29,27,24,25,23);(7,11)(10,12)(14,15)(16,24)(17,21)(18,28)(19,22)(20,27)(23,26)(25,29);(7,11)(10,12)(14,16,17,25,19,27,18,26)(15,20,21,23,22,24,28,29)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19984);