Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#12741 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1786 |
| commutator | C2×C8 | C22×C4×C5 |
| Frattini, cosocle | C22×C8 | C23×C5 |
| prime radical | C2×C5×#576 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
930 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)(8,17,19,13)(9,12)(10,24)(11,15)(14,21)(16,22)(18,30,35,25)(20,29,34,32)(23,26)(27,36)(28,33,31,37);(7,12)(9,21)(10,22)(13,25)(14,16)(15,26)(17,30)(18,33)(20,34)(23,36)(24,27)(28,29)(31,32)(35,37);(6,8)(7,13)(9,18)(10,20)(11,19)(12,25)(14,28)(15,17)(16,29)(21,33)(22,34)(23,35)(24,32)(26,30)(27,31)(36,37)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 5729);