Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C5×#1916 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C5 | C23 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle, prime radical | C24×C5 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C5 ◁ G | |
1134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(23,24,25,26,27);(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16,20,17,19,22,18,21);(8,11)(12,14)(15,17,20,22,19,21,18,16);(4,6)(7,9,10,13)(8,12,11,14)(15,18,19,20)(16,21,22,17)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14006);