Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×#9628 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #121 |
| commutator | C42 | C22×C4×C5 |
| Frattini, cosocle | C2×C42 | C23×C5 |
| socle | C22×C5 | #389 |
| prime radical | C23×C5 | C22⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7,9,12)(8,10,13,11)(14,15,19,23,20,24,18,28)(16,21,25,29,26,22,17,27);(7,10)(8,13)(11,12)(14,16,19,25,20,26,18,17)(15,21,23,29,24,22,28,27);(14,17,18,26,20,25,19,16)(15,22,23,27,24,21,28,29)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 2616);