Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C5×C8 · Schur multiplier (H2) = C24 |
| direct product | C5×#12981 |
| subgroup | G/H | |
|---|---|---|
| center | C5×C8 | #120 |
| commutator | C23 | C22×C5×C8 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1869 |
| prime radical | C5×#525 | C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C5×C8 ◁ C22×C5×C8 ◁ G | |
488 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)(6,7,10,15,11,16,24,30,12,8,13,18,25,19,26,32)(9,17,21,29,22,31,34,37,20,14,23,27,33,28,35,36);(6,8,10,18,11,19,24,32,12,7,13,15,25,16,26,30)(9,17,23,27,22,31,35,36,20,14,21,29,33,28,34,37);(6,9)(7,14)(8,17)(10,23)(11,22)(12,20)(13,21)(15,29)(16,28)(18,27)(19,31)(24,35)(25,33)(26,34)(30,37)(32,36)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5969);