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×#27018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C22×C5×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C4⋊C4 ◁ G | |
706 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)(6,7,10,13)(8,12,9,11)(14,15,17,19)(16,20,21,18)(22,23,26,31)(24,32,34,28)(25,29,27,30)(33,36,35,37);(6,8,10,9)(7,11,13,12)(14,16,17,21)(15,18,19,20)(22,24)(23,28)(25,33)(26,34)(27,35)(29,37)(30,36)(31,32);(15,19)(18,20)(22,25)(23,29)(26,27)(30,31)(33,35)(36,37);(6,9,10,8)(7,12,13,11)(15,19)(18,20)(22,25,26,27)(23,30,31,29)(24,33,34,35)(28,36,32,37)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 20006);