Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42×C5 · Schur multiplier (H2) = C22×C43 |
| direct product | C5×#13816 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator | C22 | C22×C42×C5 |
| Frattini, cosocle | C24 | C24×C5 |
| prime radical | C25×C5 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
1906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11,13,12)(15,17)(18,19,22,23)(20,24,25,21);(10,11,13,12)(19,23)(21,24);(6,7,9,8)(10,11,13,12)(14,15)(16,17)(18,20)(19,21)(22,25)(23,24);(6,8,9,7)(10,12,13,11)(18,21)(19,20)(22,24)(23,25)500 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6804);