Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13024 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #447 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| socle | C2×C5 | #1879 |
| prime radical | C5×#3414 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C2×C5×#97 ◁ G | |
466 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)(6,7,12,18,13,19,11,30,14,20,31,37,29,36,10,17)(8,21,22,34,23,35,9,25,24,16,27,15,26,32,28,33);(6,8,13,26,14,24,29,23)(7,15,19,25,20,34,36,33)(9,11,22,31,28,10,27,12)(16,17,35,18,21,30,32,37);(6,9)(7,16)(8,11)(10,24)(12,23)(13,22)(14,28)(15,17)(18,25)(19,35)(20,21)(26,31)(27,29)(30,34)(32,36)(33,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6012);