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×#11872 |
| 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 | |
372 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)(6,7,10,16,12,18,11,17)(8,19,22,14,23,32,20,33)(9,13,26,29,27,30,24,15)(21,34,36,28,25,31,37,35);(7,13)(8,20)(9,24)(10,11)(14,31)(15,17)(16,29)(18,30)(19,28)(22,23)(26,27)(32,35)(33,34)(36,37);(6,8,11,20,12,23,10,22)(7,14,16,32,18,33,17,19)(9,25,26,37,27,21,24,36)(13,28,15,34,30,35,29,31)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4860);