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×#26085 |
| 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 | |
678 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)(6,7,9,11)(8,12,13,10)(14,15,20,24)(16,22,17,26)(18,28,29,25)(19,27,21,23);(14,16,19,25,20,17,21,28)(15,18,23,22,24,29,27,26);(6,8,9,13)(7,10,11,12)(14,17,20,16)(15,22,24,26)(18,23,29,27)(19,28,21,25);(14,18,21,26,20,29,19,22)(15,17,27,25,24,16,23,28)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19073);