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×#21779 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #2308 |
| commutator | C23 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3485 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ G | |
1270 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)(6,7,10,15,11,14,23,29)(8,16,19,30,18,12,22,25)(9,13,20,28,21,27,34,37)(17,24,32,35,31,26,33,36);(6,8)(7,12)(9,17)(10,22)(11,18)(13,26)(14,16)(15,30)(19,23)(20,33)(21,31)(24,27)(25,29)(28,35)(32,34)(36,37);(6,9,11,21)(7,13,14,27)(8,17,18,31)(10,20,23,34)(12,24,16,26)(15,28,29,37)(19,32,22,33)(25,35,30,36);(7,14)(10,23)(12,16)(13,27)(19,22)(20,34)(24,26)(32,33)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 14767);