Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17629 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C5×#3506 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×D4 ◁ G | |
1010 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)(7,13,15,28)(8,18)(10,24)(14,27,31,30)(16,29,17,25)(19,34)(20,21)(32,37,33,36);(6,7)(8,13)(9,14)(10,25)(11,16)(12,17)(15,26)(18,29)(19,27)(20,30)(21,36)(22,32)(23,33)(24,28)(31,35)(34,37);(6,8,12,10)(7,13,17,25)(9,20,23,34)(11,18,26,24)(14,30,33,37)(15,28,16,29)(19,22,21,35)(27,32,36,31);(6,9,12,23)(7,14,17,33)(8,19,10,21)(11,22,26,35)(13,27,25,36)(15,31,16,32)(18,34,24,20)(28,30,29,37)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10617);