Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1408 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#7499 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #353 |
| commutator | C42 | C42×C5 |
| Frattini, cosocle | C2×C4⋊Q8 | C22×C5 |
| prime radical | C23×C5 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5 ◁ C22×C4×C5 ◁ C2×C5×C4⋊Q8 ◁ G | |
272 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 320 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5)(10,11,12,17,14,19,21,24)(13,18,22,25,23,16,20,15);(6,7,8,9)(10,12,13,20,14,21,23,22)(11,15,16,17,19,25,18,24)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 487);