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×#19409 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C5 |
| prime radical | C5×#3038 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G | |
704 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)(6,7,10,12)(8,13,11,9)(14,15)(16,20)(17,22)(18,27)(19,23)(21,24)(25,29)(26,28);(6,8,9,7,10,11,13,12)(14,16)(15,20)(17,26)(18,24)(19,25)(21,27)(22,28)(23,29);(14,16)(15,21)(17,24)(18,26)(19,25)(20,27)(22,29)(23,28);(14,17,19,18)(15,22,23,27)(16,24,25,26)(20,21,29,28)245 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 12397);