Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C2×C4 |
| direct product | C5×#12677 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #442 |
| commutator | C8 | C2×C42×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C23×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3,4,5)(10,11,13,19,14,20,23,18,15,21,25,16,24,17,12,22);(6,7,8,9)(11,16)(12,13)(14,24)(17,22)(18,20)(19,21)(23,25);(11,17,21,20)(12,23)(13,25)(16,22,19,18)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5665);