Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11271 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #121 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C23×C4×C5 ◁ G | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7,8,9)(10,11,13,12)(14,15,19,22)(16,23,28,29)(17,24,21,18)(20,27,25,26);(10,12,13,11)(14,16,24,27,19,28,18,26)(15,20,17,29,22,25,21,23);(11,12)(14,17,18,15,19,21,24,22)(16,25,26,29,28,20,27,23)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4259);