Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#11356 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C5 | #390 |
| commutator | C22×C4 | C22×C4×C5 |
| Frattini, cosocle | C23×C4 | C23×C5 |
| socle | C2×C5 | #1445 |
| prime radical | C24×C5 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C5 ◁ C23×C4×C5 ◁ G | |
580 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)(8,16)(9,20,22,13)(10,23)(11,15)(12,18)(14,25)(17,31,33,37)(19,30)(21,29,36,34)(24,27)(26,35,28,32);(6,8,10,24,11,19,25,18)(7,12,23,16,15,27,14,30)(9,17,21,35,22,33,36,32)(13,28,34,37,20,26,29,31);(6,9,10,21,11,22,25,36)(7,13,23,34,15,20,14,29)(8,17,24,35,19,33,18,32)(12,26,16,31,27,28,30,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 4344);