Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5×C8 · Schur multiplier (H2) = C26 |
| direct product | C5×#33337 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C22×C22⋊C4 |
| commutator | C22 | C23×C5×C8 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C22×C22⋊C8 |
| prime radical | C22×C5×#41 | C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C2×C5×#131 ◁ G | |
1128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,9,14,10,15,20,27,11,12,19,23,21,24,31,34)(8,13,17,25,18,26,30,35,16,22,28,32,29,33,36,37);(7,12)(9,19)(13,22)(15,24)(17,28)(20,31)(26,33)(30,36);(6,8)(7,13)(9,17)(10,18)(11,16)(12,22)(14,25)(15,26)(19,28)(20,30)(21,29)(23,32)(24,33)(27,35)(31,36)(34,37);(8,16)(13,22)(17,28)(18,29)(25,32)(26,33)(30,36)(35,37)410 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26325);