Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8794 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #382 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,7)(3,8)(4,5)(9,10)(11,19,23,15)(12,24,27,38)(13,18)(14,17)(16,26,32,35)(20,28)(21,36)(22,33,34,30)(25,39)(29,37)(31,40);(1,2,4,7,6,8,5,3)(9,11,20,34,14,23,13,22)(10,15,28,33,17,19,18,30)(12,25,35,37,27,21,26,40)(16,31,38,39,32,29,24,36);(1,3,4,2,6,7,5,8)(9,12)(10,16,17,32)(11,21)(13,26)(14,27)(15,29,19,31)(18,24,28,38)(20,35)(22,37)(23,25)(30,39,33,36)(34,40)Polycyclic presentation:
2^8|a^2=f,b^2=eg,c^2=d,d^2=h,e^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,f^b=fgh,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#8795, #191228, #191230, … |
| maximal normal subgroup of | C2×#8795, #191228, #191230, … |
| maximal quotient of | #99282, #99289, #108607, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1783);