Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#7431 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | AΣL(1,𝔽16) |
| commutator | C2×C4⋊C4 | C2×C4×C5 |
| Frattini, cosocle | #391 | C22×C5 |
| prime radical | C23×C5 | #99 |
| LCS | G ▷ C2×C4⋊C4 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5 ◁ C22×C4×C5 ◁ C5×#391 ◁ G | |
206 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5)(7,10,11,12)(14,15,18,22,20,16,24,25)(17,27,19,23,28,29,21,26);(6,7)(8,12)(9,11)(10,13)(14,16,19,26,20,15,21,23)(17,25,24,29,28,22,18,27)155 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 419);