Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#23015 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | C22×D4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×C4×D4 |
| prime radical | C5×#3035 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C5×C4⋊C8 ◁ G | |
892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2,3,4,5)(6,7,9,8)(10,11,14,18,15,13,22,17)(12,16,20,24,21,19,25,23);(7,8)(10,12,14,20,15,21,22,25)(11,16,18,24,13,19,17,23);(10,13)(11,15)(12,19)(14,17)(16,21)(18,22)(20,23)(24,25);(6,8,9,7)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16003);