Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7035 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#7036 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #95 |
| commutator | C22×C4 | C42×C5 |
| Frattini, cosocle | C22×C42 | C22×C5 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C22×C42×C5 ◁ G | |
458 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 458 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 320 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5)(6,7,9,12,10,13,8,11)(14,15)(16,20)(17,19)(18,21)(23,26,27,28);(7,11,13,12)(14,16)(15,18,19,20)(17,21)(22,23)(24,28)(25,27)(26,29)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 24);