Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23 |
| direct product | C5×#26019 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C24×C5 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C22⋊C4 ◁ G | |
696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 37 points:
(1,2,3,4,5)(6,7)(8,12)(9,11)(10,13)(14,15,17,19)(16,20,21,18)(22,23)(24,28)(25,30)(26,29)(27,31)(32,37)(33,35)(34,36);(6,8,9,13)(7,10,11,12)(22,24,26,34)(23,28,29,36)(25,33,27,32)(30,35,31,37);(14,16,17,21)(15,18,19,20)(23,29)(24,32)(25,27)(28,35)(33,34)(36,37);(7,11)(10,12)(22,25,26,27)(23,30,29,31)(24,33,34,32)(28,35,36,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 19007);