Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#33530 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×#447 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×M16 ◁ G | |
1132 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)(8,17)(9,14)(10,22)(11,26)(12,16)(13,19)(15,25)(18,34)(20,33)(21,32)(23,28)(24,36)(27,31)(29,35)(30,37);(6,8)(7,13)(9,18)(10,23)(11,27)(12,20)(14,28)(15,31)(16,29)(17,22)(19,25)(21,35)(24,37)(26,30)(32,34)(33,36);(10,24)(15,32)(17,33)(19,35)(21,25)(22,36)(23,37)(31,34);(6,9,12,11)(7,14,16,26)(8,18,20,27)(10,21,24,25)(13,28,29,30)(15,22,32,36)(17,34,33,31)(19,23,35,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26518);