Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10603 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C2 |
| direct product | C5×#10605 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | #396 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C23×C5 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,3,4,5)(6,7,8,13,10,9,11,12)(14,15,19,21,18,22,17,26)(16,23,27,28,24,29,25,20);(6,8,10,11)(7,12,9,13)(14,16,17,25,18,24,19,27)(15,20,21,23,22,28,26,29);(6,9,10,7)(8,13,11,12)(16,24)(17,19)(21,26)(23,29)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3593);