Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C27 |
| direct product | C2×C5×#2528 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C5 | C24 |
| commutator, Frattini, cosocle | C23 | C25×C5 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C5 ◁ G | |
1878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 5 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4,6,8)(5,7,9,10)(13,17)(16,18)(20,24)(22,26);(3,4,6,8)(5,7,9,10)(11,12,15,14)(13,16,17,18)(19,20,23,24)(21,22,25,26);(3,4)(5,7)(6,8)(9,10)(11,13)(12,16)(14,18)(15,17)(19,21,23,25)(20,22,24,26);(3,5,6,9)(4,7,8,10)(11,14,15,12)(13,18,17,16)(19,22)(20,25)(21,24)(23,26)380 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 27780);