Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×C5×#1772 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C5×C8 | D4 |
| commutator | C4 | C22×C42×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C23×C5 | C4×D4 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5×C8 ◁ C23×C5×C8 ◁ G | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 27 points:
(1,2)(23,24,25,26,27);(8,12,14,17)(13,19,20,21);(3,4,5,6)(7,8)(9,13)(10,17)(11,14)(12,18)(15,21)(16,20)(19,22);(7,9,10,15,11,16,18,22)(8,13,17,21,14,20,12,19)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 13430);