Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C27 |
| direct product | C2×C5×#2029 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #120 |
| commutator, Frattini, cosocle | C24 | C24×C5 |
| prime radical | C26×C5 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C25×C5 ◁ G | |
2818 conjugacy classes of subgroups; only the 350 normal subgroups are shown.
This group has 2818 conjugacy classes of subgroups — too many to draw. Only its 350 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(3,4,6,9)(5,11,12,8)(7,10,14,16)(13,17,18,15);(3,5,6,12)(4,8,9,11)(7,13,14,18)(10,15,16,17);(8,15)(11,17)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 16453);