Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#7491 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C42×C5 |
| Frattini, cosocle | C2×C4⋊Q8 | C22×C5 |
| prime radical | C23×C5 | #99 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5 ◁ C22×C4×C5 ◁ C2×C5×C4⋊Q8 ◁ G | |
276 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 276 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5)(10,11,14,18,15,12,13,21)(16,19,23,20,25,22,24,17);(6,7,8,9)(10,12,15,11)(13,17,14,20)(16,19,25,22)(18,24,21,23)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 479);