Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33348 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#99 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C2×D4 | C24×C5 |
| prime radical | C5×#3484 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×D4 ◁ C2×C5×H(5,𝔽2) ◁ G | |
1410 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(6,7)(8,12)(9,16,17,11)(10,13)(14,19)(15,21,20,18);(7,11)(12,18)(13,16)(19,21);(6,8)(7,12)(9,15)(10,14)(11,18)(13,19)(16,21)(17,20);(8,14)(12,19)(15,20)(18,21)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26336);