Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18562 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18611 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3497 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
1106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(6,7,10,13)(8,12,9,11)(14,15)(16,23)(17,21)(18,26)(19,22)(20,24)(25,28)(27,29);(7,11)(8,9)(12,13)(14,15)(16,20)(17,21)(18,26)(19,22)(23,27)(24,28)(25,29);(14,16,19,25)(15,20,22,29)(17,24,18,27)(21,28,26,23);(6,8,10,9)(7,12,13,11)(14,17,19,18)(15,21,22,26)(16,24,25,27)(20,28,29,23)200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11599);