Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #21776 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C26 |
| direct product | C5×#21781 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2308 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3487 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7,11,17,12,16,9,14)(8,18,19,27,21,13,25,29)(10,23,22,31,24,15,26,30)(20,28,35,36,34,32,33,37);(6,8)(7,13)(9,19)(10,20)(11,25)(12,21)(14,29)(15,28)(16,18)(17,27)(22,33)(23,32)(24,34)(26,35)(30,37)(31,36);(6,9,12,11)(7,14,16,17)(8,19,21,25)(10,22,24,26)(13,27,18,29)(15,30,23,31)(20,33,34,35)(28,36,32,37);(6,10,12,24)(7,15,16,23)(8,20,21,34)(9,22,11,26)(13,28,18,32)(14,30,17,31)(19,33,25,35)(27,36,29,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14769);