Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C29 |
| direct product | C5×#32898 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C25×C5 ◁ G | |
4360 conjugacy classes of subgroups; only the 370 normal subgroups are shown.
This group has 4360 conjugacy classes of subgroups — too many to draw. Only its 370 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) = 160 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,2,3,4,5)(6,7)(8,9)(14,17);(10,11)(12,14)(13,15)(16,17);(10,12)(11,14)(13,16)(15,17);(7,9)245 conjugacy classes — view the full character table.
| Fitting subgroup of | D5×#32898, AGL(1,𝔽5)×#32898 |
| maximal subgroup of | D5×#32898, C5×#11715850 |
| maximal normal subgroup of | D5×#32898 |
| maximal quotient of | C5×#511790, C5×#347042 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 25886);