Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13323 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#13323 |
| 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 | |
534 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)(7,13)(8,18)(9,22)(10,11)(14,31)(15,29)(16,23)(17,30)(19,32)(20,24)(21,27)(25,26)(28,33)(34,37);(6,7)(8,19)(9,23)(10,15)(11,16)(12,17)(13,22)(14,18)(20,28)(21,33)(24,32)(25,30)(26,29)(27,34)(31,35)(36,37);(6,8)(7,14)(9,24)(10,27)(11,18)(12,21)(13,28)(15,19)(16,33)(17,34)(20,26)(22,36)(23,37)(25,35)(29,31)(30,32)170 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 6311);