Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C4×C5×#356 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #99 |
| commutator | C2×C4 | C2×C42×C5 |
| Frattini, cosocle | C22×Q8 | C23×C5 |
| socle | C22×C5 | C2×#99 |
| prime radical | C23×C5 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×Q8 ◁ G | |
500 conjugacy classes of subgroups; only the 166 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 166 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 — 3 generators on 25 points:
(1,2,3,4)(21,22,23,24,25);(5,6,9,13,10,14,7,15)(8,12,16,19,18,20,17,11);(5,7,10,9)(6,11,14,19)(8,17,18,16)(12,13,20,15)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5618);