Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7307 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#7308 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #342 |
| commutator | C22×C4 | C42×C5 |
| Frattini, cosocle | C23×Q8 | C22×C5 |
| prime radical | C24×C5 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C24×C5 ◁ C23×C5×Q8 ◁ G | |
518 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 518 conjugacy classes of subgroups — too many to draw. Only its 86 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)(9,11)(10,13)(14,15,17,21,18,22,26,29)(16,20,23,28,24,19,25,27);(6,8)(7,10,11,12)(9,13)(15,19,22,20)(21,28,29,27)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 296);