Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C5 · Schur multiplier (H2) = C28 |
| direct product | C5×#61318 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C5 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C5 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C5 ◁ C2×C4×C5×Q8 ◁ G | |
1880 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 5 generators on 29 points:
(1,2,3,4,5)(7,10)(11,12)(14,15)(16,19)(17,20)(18,21)(22,25)(23,26)(24,27)(28,29);(6,7,9,10)(8,12,13,11)(14,16,18,24)(15,19,21,27)(17,23,25,29)(20,26,22,28);(15,20)(17,25)(19,26)(21,22)(23,29)(27,28);(6,8,9,13)(7,11,10,12);(15,21)(19,27)(20,22)(26,28)275 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 54306);