Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12826 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#12826 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1804 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C5×#405 ◁ G | |
412 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2,3,4,5)(6,7)(8,18,21,35)(9,15)(10,14)(11,27)(12,17)(13,19,30,22)(16,25)(20,33,24,29)(23,31)(26,32)(28,37,34,36);(6,8)(7,13,17,30)(9,19)(10,24)(11,22)(12,21)(14,28,32,34)(15,18,27,35)(16,33,31,29)(20,26)(23,37)(25,36);(7,14)(9,11)(10,25)(13,28)(15,16)(17,32)(18,33)(19,22)(20,37)(23,26)(24,36)(27,31)(29,35)(30,34)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5814);