Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7453 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C5 · Schur multiplier (H2) = C1 |
| direct product | C5×#7453 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1365 |
| commutator | C2×C16 | C2×C4×C5 |
| Frattini, cosocle | C4×C16 | C22×C5 |
| prime radical | C22×C5 | #361 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C4×C5×C8 ◁ C4×C5×C16 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 2 generators on 37 points:
(1,2,3,4,5)(6,7,9,16,11,29,8,20,10,28,21,32,22,37,23,15,13,19,27,14,31,18,25,36,12,17,26,35,30,33,24,34);(7,14,19,16)(8,21)(9,23)(10,12)(11,30)(15,18,34,29)(17,35,28,32)(20,33,36,37)(22,31)(24,27)(25,26)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 441);