Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C5 · Schur multiplier (H2) = C25 |
| direct product | C2×C5×#2224 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C5 |
| prime radical | C22×C5×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ C2×C4×C5×C8 ◁ G | |
638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2)(27,28,29,30,31);(3,4,6,8)(5,9,10,7)(11,12)(13,16)(14,23)(15,19)(17,20)(18,21)(22,25)(24,26);(3,5,6,10)(4,7,8,9);(12,16)(13,20)(14,21)(17,18)(19,25)(22,26)(23,24)290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 26608);