Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#7307 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #342 |
| commutator | C22×C4 | C42×C5 |
| Frattini, cosocle | C23×D4 | C22×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C24×C5 ◁ C23×C5×D4 ◁ G | |
794 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 794 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 2 generators on 21 points:
(1,2,3,4,5)(6,7)(8,12,17,11)(9,15)(10,14)(13,18)(16,19,21,20);(7,11,13,19)(8,16)(12,14,20,15)(17,21)170 conjugacy classes — view the full character table.
| Fitting subgroup of | D5×#7307, AGL(1,𝔽5)×#7307, #72242861, … |
| maximal subgroup of | D5×#7307, #72242897, C5×#144366 |
| maximal normal subgroup of | D5×#7307, #72242897, C5×#144366 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 295);