Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C22×C5×#352 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #99 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C2×D4 | C24×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C24×C5 ◁ C23×C5×D4 ◁ G | |
1486 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)(21,22,23,24,25);(3,4);(5,6)(7,13)(8,16,17,11)(9,12)(10,14,18,20)(15,19);(5,7,9,15)(6,10,12,18)(8,14,17,20)(11,13,16,19)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26330);