Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C5×C8 · Schur multiplier (H2) = C4 |
| direct product | C5×#7357 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #353 |
| commutator | C42 | C2×C5×C8 |
| Frattini, cosocle | #423 | C22×C5 |
| socle | C2×C5 | #1318 |
| prime radical | C5×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C5×#423 ◁ G | |
222 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 222 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) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,7,9,13,11,14,18,20,12,16,19,21,10,15,8,17);(6,8)(7,13)(9,11)(10,19)(12,18)(14,20)(15,17)(16,21)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 345);