Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C16 · Schur multiplier (H2) = C23 |
| direct product | C5×C16×D8 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5×C16 | D4 |
| commutator | C4 | C22×C5×C16 |
| Frattini, cosocle | C4×C8 | C23×C5 |
| socle | C22×C5 | C8×D4 |
| prime radical | C2×C5×D8 | C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5×C16 ◁ C4×C5×C16 ◁ G | |
258 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21);(22,23,24,25,26,27,28,29);(23,29)(24,28)(25,27)560 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5025);