Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2613 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C2×C5×#2644 |
| 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 | |
2222 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);(4,7)(8,9)(11,12)(13,16)(14,17)(15,18)(19,20,23,25)(21,26,22,24);(3,4,6,7)(5,8,10,9)(11,13)(12,14)(15,17)(16,18)(19,21,23,22)(20,24,25,26);(3,4)(5,9)(6,7)(8,10)(11,13)(12,16)(14,18)(15,17)(19,20)(21,24)(22,26)(23,25);(3,5,6,10)(4,8,7,9)(11,14,15,16)(12,17,18,13)(19,22,23,21)(20,24,25,26)320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27605);