Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C5×C8×C16 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7136 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5×C8 | C22 |
| commutator | C2 | C5×C8×C16 |
| Frattini, cosocle | C2×C4×C8 | C22×C5 |
| socle, prime radical | C23×C5 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5×C8 ◁ G | |
306 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,2,3,4,5)(6,7)(8,9)(18,19,20,23,21,24,26,29,22,25,27,30,28,31,32,33);(7,9)(10,11,12,14,13,15,16,17)800 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 124);