Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C4 |
| direct product | C5×#13733 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C5 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C23×C5 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C42×C5 ◁ C4×C5×C8 ◁ C4×C5×C16 ◁ G | |
158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 45 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 3 generators on 45 points:
(1,2,3,4,5)(6,7,9,11)(8,12,13,10)(15,20)(16,25)(17,26)(18,29)(21,28)(22,37)(23,40)(24,34)(27,36)(30,43)(31,32)(33,38)(35,44)(39,45)(41,42);(6,8,9,13)(7,10,11,12);(14,15,21,35,17,22,27,33,29,38,42,37,43,44,16,20,19,24,25,32,30,39,41,40,18,23,36,45,26,31,28,34)350 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6721);