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×#26401 |
| 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 | |
714 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)(6,7,10,12)(8,11,9,13)(14,15)(16,23)(17,19)(18,22)(20,24)(21,27)(25,28)(26,29);(6,8,10,9)(7,11,12,13)(15,19)(17,27)(20,23)(21,22)(24,25)(28,29);(6,9,10,8)(7,11,12,13)(14,15)(16,20)(17,21)(18,22)(19,27)(23,24)(25,29)(26,28);(14,16,18,26)(15,20,22,28)(17,25,27,24)(19,23,21,29)215 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19389);