Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8608 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#8615 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #380 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(14,15,17,21,18,20,26,28)(16,22,23,29,24,19,25,27);(6,7,8,10,9,11,12,13);(14,16,18,24)(15,19,20,22)(17,25,26,23)(21,29,28,27)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 1603);