Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29610 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C24×C4 |
| direct product | C5×#29619 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G | |
1332 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)(8,11)(9,10)(12,13)(14,15,17,21)(16,23,24,29)(18,27,19,22)(20,26,28,25);(7,10)(11,13)(14,15)(16,23)(17,21)(18,22)(19,27)(20,26)(24,29)(25,28);(6,8)(7,11)(9,12)(10,13)(14,16,17,24)(15,20,21,28)(18,26,19,25)(22,29,27,23);(8,12)(11,13)(16,24)(20,28)(23,29)(25,26)185 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 22607);