Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#12687 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#99 |
| commutator | C24 | C22×C4×C5 |
| Frattini, cosocle | C22×D4 | C23×C5 |
| prime radical | C22×C5×#120 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C24×C5 ◁ C23×C5×D4 ◁ G | |
2508 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 2508 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3,4,5)(6,7)(8,9)(10,11)(12,16,17,14)(13,15);(11,14)(15,16);(7,9)200 conjugacy classes — view the full character table.
| Fitting subgroup of | D5×#12687, AGL(1,𝔽5)×#12687, #72242830, … |
| maximal subgroup of | D5×#12687, #72242893, C5×#144366, … |
| maximal normal subgroup of | D5×#12687, #72242893, C5×#144366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5675);