Finite Group DatabasebetaFinite Group Databasebeta| order | 1280 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10046 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C5 · Schur multiplier (H2) = C24 |
| direct product | C5×#10063 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C5 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23×C5 |
| prime radical | C2×C5×#127 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G | |
670 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 640 |
A minimal faithful permutation representation — 3 generators on 33 points:
(1,2,3,4,5)(10,11)(12,16)(13,15)(14,17)(18,19,22,25)(20,26,27,33)(21,29,30,24)(23,31,32,28);(6,7)(8,9)(11,15)(12,13)(16,17)(21,30)(24,29)(26,33)(28,31);(7,9)(10,12,14,13)(11,16,17,15)(18,20)(19,23)(21,31)(22,27)(24,33)(25,32)(26,29)(28,30)230 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1280, 3051);