Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C23×C4 |
| direct product | C5×#19620 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #127 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| socle | C22×C5 | C2×#127 |
| prime radical | C5×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
712 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)(8,14,15,21)(9,17,18,12)(11,19,20,16)(22,23)(24,31)(25,29)(26,35)(27,30)(28,33)(32,36)(34,37);(6,8,9,16,10,15,18,19)(7,11,12,14,13,20,17,21);(8,15)(11,20)(14,21)(16,19)(22,24,26,33,27,34,25,32)(23,28,29,31,30,36,35,37);(22,25,27,26)(23,29,30,35)(24,32,34,33)(28,31,36,37)260 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12608);