Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#12573 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #99 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C22×D4 | C23×C5 |
| socle | C22×C5 | C2×#99 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C22×C5×#41 ◁ G | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(10,11)(12,16)(13,20,21,15)(14,17)(18,24,25,22)(19,23);(6,7,8,9)(11,15)(16,22)(17,20)(23,24);(10,12,14,19)(11,16,17,23)(13,18,21,25)(15,22,20,24)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5561);