Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7606 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#7638 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C5 | C2×C22⋊C4 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C43×C5 ◁ G | |
468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(7,11,12,13)(14,15,17,19,18,20,16,21)(22,23,26,29,27,24,25,28);(6,7)(8,11)(9,13)(10,12)(15,19)(16,17)(20,21)(22,24,27,23)(25,29,26,28);(6,8,10,9)(7,11,12,13)(15,20)(19,21)(22,25,27,26)(23,28,24,29)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 626);