Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#33594 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #459 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×#459 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
696 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)(9,22)(10,16)(11,26)(12,27)(13,18)(14,20)(15,19)(21,34)(23,37)(24,33)(25,29)(28,30)(31,35)(32,36);(6,8,11,20,13,21,10,12)(7,14,16,17,18,27,26,34)(9,19,24,35,25,36,23,28)(15,29,30,33,32,22,31,37);(6,9)(7,15)(8,19)(10,23)(11,24)(12,28)(13,25)(14,29)(16,30)(17,33)(18,32)(20,35)(21,36)(22,27)(26,31)(34,37);(6,10,13,11)(7,16,18,26)(8,12,21,20)(9,23,25,24)(14,17,27,34)(15,30,32,31)(19,28,36,35)(22,37,29,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26582);