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×#25174 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C5 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C4×C5×C22⋊C4 ◁ G | |
1234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,3,4,5)(10,11,13,12)(15,19)(16,17)(20,21);(6,7)(8,9)(10,12,13,11)(14,15,18,20)(16,19,17,21);(7,9)(11,12)(14,16,18,17)(15,19,20,21);(14,15)(16,19)(17,21)(18,20)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 18162);