Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C22×C4 |
| direct product | C5×#7504 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #354 |
| commutator | C42 | C42×C5 |
| Frattini, cosocle | C2×#127 | C22×C5 |
| prime radical | C2×C5×#127 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C5 ◁ C22×C4×C5 ◁ C2×C5×#127 ◁ G | |
336 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 25 points:
(1,2,3,4,5)(10,11)(12,19,21,24)(13,17,22,16)(14,18)(15,20,23,25);(6,7,8,9)(10,12,13,21,14,20,22,25)(11,15,17,19,18,24,16,23)140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 492);