Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C23 |
| direct product | C5×#11485 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #99 |
| commutator | C2×C4 | C22×C5×C8 |
| Frattini, cosocle | C2×M16 | C23×C5 |
| socle | C2×C5 | #1459 |
| prime radical | C5×#532 | C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C2×C5×C8 ◁ C2×C5×#131 ◁ G | |
372 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7,11,17,12,16,28,34,13,18,29,36,10,26,20,35)(8,19,22,37,21,14,9,15,23,31,24,32,27,30,25,33);(6,8)(7,14)(9,20)(10,27)(11,24)(12,21)(13,23)(15,17)(16,31)(18,30)(19,26)(22,29)(25,28)(32,34)(33,36)(35,37);(6,9,12,24,13,25,10,22)(7,15,16,32,18,33,26,37)(8,20,21,11,23,28,27,29)(14,17,31,34,30,36,19,35)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4473);