Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#7376 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C5×C8 |
| Frattini, cosocle | #391 | 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 ◁ C5×#391 ◁ G | |
202 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 202 conjugacy classes of subgroups — too many to draw. Only its 50 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)(6,7,8,10,9,11,12,13)(14,15,17,20)(16,23,25,29)(18,21,27,28)(19,22,26,24);(14,15,18,21,19,22,25,29)(16,24,26,28,27,20,17,23)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 364);