Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#19072 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
992 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(14,15)(16,24)(17,25)(18,23)(19,22)(20,21)(26,29)(27,28);(8,12)(10,13)(14,16)(15,17)(18,28)(19,25)(20,26)(21,27)(22,29)(23,24);(6,7)(8,13)(9,11)(10,12)(14,17)(15,16)(18,24)(19,29)(20,27)(21,26)(22,25)(23,28);(15,21)(17,27)(22,23)(24,29)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12060);