Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C26 |
| direct product | C22×C5×Q8⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C5 | D4 |
| commutator | C4 | C23×C5×C8 |
| Frattini, cosocle | C42 | C24×C5 |
| socle, prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C5 ◁ C22×C42×C5 ◁ G | |
1454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(21,22,23,24,25);(3,4);(6,9)(7,11)(10,12)(13,14,15,17,16,18,19,20);(5,6,8,10)(7,12,11,9)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10420);