Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C5×#25178 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| prime radical | C2×C5×#449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C22⋊C4 ◁ G | |
1134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 29 points:
(1,2,3,4,5)(10,11,13,12)(14,15)(16,24)(17,22)(18,20)(19,23)(21,27)(25,28)(26,29);(6,7)(8,9)(10,12,13,11)(15,20)(17,18)(21,28)(22,23)(24,29)(25,26);(7,9)(11,12)(14,16,19,27)(15,21,23,24)(17,25,18,26)(20,28,22,29);(14,17)(16,25)(18,19)(20,22)(26,27)(28,29)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18166);