Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5×C8 · Schur multiplier (H2) = C29 |
| direct product | C5×C8×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C8 | C24 |
| commutator | C2 | C24×C5×C8 |
| Frattini, cosocle | C2×C4 | C25×C5 |
| socle | C22×C5 | C24×C4 |
| prime radical | C2×C5×H(5,𝔽2) | C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C8 ◁ G | |
2422 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13);(14,15)(16,20)(17,18)(19,21);(15,18)(19,20);(14,15)(16,19)(17,18)(20,21);(14,16)(15,19)(17,21)(18,20)680 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 53291);