Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33388 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33388 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | C22×#102 |
| commutator | C8 | C23×C4×C5 |
| Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C5×#3584 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C5×#41 ◁ 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)(7,12,14,26)(9,19)(10,22)(13,25,28,36)(15,20,23,27)(17,32)(18,31)(21,24)(29,33,34,37)(30,35);(6,7)(8,13)(9,20)(10,23)(11,14)(12,24)(15,22)(16,28)(17,33)(18,34)(19,26)(21,27)(25,35)(29,31)(30,37)(32,36);(6,8)(7,13)(9,17)(10,18)(11,16)(12,25)(14,28)(15,29)(19,32)(20,33)(21,30)(22,31)(23,34)(24,35)(26,36)(27,37);(8,16)(13,28)(17,30)(18,31)(25,36)(29,34)(32,35)(33,37)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26376);