Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5×C8 · Schur multiplier (H2) = C26 |
| direct product | C22×C5×#346 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C5×C8 | C22 |
| commutator | C2 | C22×C4×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle, prime radical | C24×C5 | C2×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5×C8 ◁ G | |
942 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2)(25,26,27,28,29);(3,4);(9,10,11,15,12,16,17,22,13,14,18,20,19,21,23,24);(5,6,7,8)(10,14)(15,20)(16,21)(22,24)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26309);