Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2121 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#12579 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | C2×#99 |
| commutator | C23 | C2×C42×C5 |
| Frattini, cosocle | C2×#41 | C23×C5 |
| socle | C2×C5 | C2×#342 |
| prime radical | C22×C5×#41 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C22×C5×#41 ◁ G | |
736 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)(6,7,10,15,12,17,11,16)(8,18,21,34,19,33,22,35)(9,23,24,32,25,14,26,30)(13,27,29,20,28,37,31,36);(6,8,11,22,12,19,10,21)(7,13,16,29,17,28,15,31)(9,20,24,37,25,36,26,27)(14,18,30,35,23,33,32,34);(8,19)(13,28)(18,33)(20,36)(21,22)(27,37)(29,31)(34,35)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5567);