Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11521 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#11521 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #410 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×M16 | C23×C5 |
| socle | C2×C5 | #1464 |
| prime radical | C24×C5 | M16 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C42×C5 ◁ C4×C5×M16 ◁ G | |
330 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5)(6,7,8,13,9,14,16,21,10,11,15,18,17,12,19,20);(7,11)(12,14);(7,12,11,14)(8,15)(13,21,18,20)(16,19)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4509);