Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7852 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42×C5 | C23 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C23×C5 | C2×C42 |
| prime radical | C24×C5 | C42 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C42×C5 ◁ G | |
802 conjugacy classes of subgroups; only the 334 normal subgroups are shown.
This group has 802 conjugacy classes of subgroups — too many to draw. Only its 334 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(6,7)(8,9)(10,11,12,15,13,14,16,17);(7,9)(18,19,20,23,21,22,24,25);(11,14)(15,17)(19,22)(23,25)440 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 840);