Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#8453 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #379 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| prime radical | C5×#3484 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1330 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1330 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) = 21 |
| transitive degree | μt(G) = 80 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3,4,5)(7,11,13,19)(8,15)(12,16,14,20)(17,21);(6,7)(8,16)(9,14)(10,12)(11,17)(13,18)(15,19)(20,21);(8,17)(11,16)(15,21)(19,20)200 conjugacy classes — view the full character table.
| Fitting subgroup of | D5×#8453, #72242841, AGL(1,𝔽5)×#8453, … |
| maximal subgroup of | D5×#8453, #72242895, C5×#144366 |
| maximal normal subgroup of | D5×#8453, #72242895, C5×#144366 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1441);