Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#19390 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C22×C5×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
674 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)(14,15,18,22)(16,19,23,27)(17,20,25,21)(24,26,29,28);(6,7,8,10,9,11,12,13)(14,16)(15,19)(17,24)(18,23)(20,26)(21,28)(22,27)(25,29);(15,20)(17,25)(19,26)(21,22)(24,29)(27,28);(16,23)(19,27)(24,29)(26,28)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12378);