Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13318 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13318 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C5 | #1999 |
| commutator, Frattini, cosocle | C22×C8 | C23×C5 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C4×C5 ◁ C2×C5×#131 ◁ G | |
532 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)(8,19)(9,23)(10,18)(11,16)(12,17)(13,20)(14,21)(15,27)(22,35)(24,36)(25,31)(26,32)(28,34)(29,33)(30,37);(6,8)(7,13)(9,24)(10,21)(11,29)(12,22)(14,32)(15,33)(16,30)(17,31)(18,36)(19,23)(20,27)(25,26)(28,37)(34,35);(6,8)(7,14)(9,25)(10,29)(11,21)(12,22)(13,15)(16,35)(17,33)(18,19)(20,28)(23,30)(24,26)(27,37)(31,32)(34,36)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6306);