Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11867 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#11867 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1519 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C2×M16 | C23×C5 |
| prime radical | C5×#3060 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C2×C5×#131 ◁ G | |
376 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)(9,22,26,12)(10,15)(11,17)(13,19)(14,23,28,36)(16,25)(20,30,24,29)(21,31)(27,37,33,34)(32,35);(7,12)(8,19)(9,23)(10,25)(13,29)(14,16)(15,28)(17,22)(18,33)(20,24)(21,35)(26,36)(27,31)(30,32);(6,8,10,19,11,21,25,35)(7,13,16,18,17,32,15,31)(9,24,23,37,26,20,36,34)(12,27,28,30,22,33,14,29)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4855);