Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17597 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#17684 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4×C5 |
| Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3517 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C4×C5×#41 ◁ G | |
932 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)(6,7,11,16,12,15,10,14)(8,17,18,21,9,13,19,20)(22,23)(24,30,32,37)(25,29)(26,28)(27,31,35,33)(34,36);(6,8,12,9)(7,13,15,17)(10,18,11,19)(14,20,16,21)(22,24)(23,27)(25,33)(26,32)(28,35)(29,37)(30,36)(31,34);(6,9,12,8)(7,13,15,17)(10,18,11,19)(14,21,16,20)(23,28)(27,35)(29,36)(30,37);(6,10,12,11)(7,14,15,16)(8,18,9,19)(13,20,17,21)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 10672);