Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C8 |
| direct product | C2×C5×#1705 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C2×C5×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
994 conjugacy classes of subgroups; only the 382 normal subgroups are shown.
This group has 994 conjugacy classes of subgroups — too many to draw. Only its 382 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) = 640 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2)(19,20,21,22,23);(4,7)(5,9)(8,10)(12,15)(13,17)(16,18);(3,4,5,7,6,8,9,10);(11,12,13,15,14,16,17,18)380 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C5×#19745, AGL(1,𝔽11)×#1705 |
| maximal normal subgroup of | C2×C5×#19745 |
| maximal quotient of | AGL(1,𝔽11)×#1705 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12580);