Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#18124 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C5 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| socle | C22×C5 | C2×#120 |
| prime radical | C5×#392 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 160 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(7,10)(8,12)(11,13)(19,25)(26,29);(6,7,9,11)(8,13,12,10)(14,15)(16,19)(17,20)(18,21)(22,26)(23,25)(24,27)(28,29);(14,16)(15,19)(17,22)(18,23)(20,26)(21,25)(24,28)(27,29);(14,17,18,24)(15,20,21,27)(16,22,23,28)(19,26,25,29)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 11112);