Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#33637 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #476 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| socle | C2×C5 | C2×#476 |
| prime radical | C2×C5×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C4×C5×C8 ◁ G | |
524 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)(8,20)(9,24,27,37)(10,16)(11,18)(12,28)(13,19)(14,22)(15,21,31,36)(17,23)(25,34,26,32)(29,33,35,30);(6,8,11,22,13,23,10,12)(7,14,16,20,19,28,18,17)(9,21,26,35,27,36,25,29)(15,24,30,34,31,37,33,32);(6,9)(7,15)(8,21)(10,25)(11,26)(12,29)(13,27)(14,24)(16,30)(17,32)(18,33)(19,31)(20,34)(22,35)(23,36)(28,37);(6,10,13,11)(7,16,19,18)(8,12,23,22)(9,25,27,26)(14,20,28,17)(15,30,31,33)(21,29,36,35)(24,34,37,32)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26625);