Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C210 |
| direct product | C2×C5×#2614 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| prime radical | C23×C5×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
5194 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(27,28,29,30,31);(8,10)(11,12,15,18)(13,19,21,16)(14,17,23,22)(20,25,26,24);(4,6)(8,10)(11,13,15,21)(12,16,18,19)(14,20,23,26)(17,24,22,25);(3,4)(5,6)(7,8)(9,10)(11,12)(13,19)(14,22)(15,18)(16,21)(17,23)(20,24)(25,26);(11,14)(12,17)(13,20)(15,23)(16,24)(18,22)(19,25)(21,26)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27586);