Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C5 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×C5×#2369 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | C24 |
| commutator | C22 | C24×C4×C5 |
| Frattini, cosocle | C23 | C25×C5 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
2726 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 5 generators on 27 points:
(1,2)(23,24,25,26,27);(3,4,6,5)(7,8,11,10)(9,13,14,12)(16,19)(20,21);(3,5,6,4)(8,10)(12,13)(15,16)(17,21)(18,19)(20,22);(7,9,11,14)(8,12,10,13);(7,10,11,8)(9,13,14,12)(15,17,18,22)(16,20,19,21)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27131);