Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C5×C8 · Schur multiplier (H2) = C22 |
| direct product | C5×#7184 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #95 |
| commutator | C2×C4 | C4×C5×C8 |
| Frattini, cosocle | C23×C8 | C22×C5 |
| socle | C22×C5 | #336 |
| prime radical | C24×C5 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C23×C5×C8 ◁ G | |
370 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3,4,5)(6,7)(8,12,13,10)(9,11)(14,15,17,22,19,21,26,28,18,23,25,29,16,20,24,27);(6,8)(7,10)(9,13)(11,12)(15,20,23,21)(22,27,29,28)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 172);