Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10440 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #123 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
204 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,14,18)(11,19,22,34)(12,23,26,16)(13,27,29,17)(15,28,31,21)(20,36,24,35)(25,33,40,38)(30,37,32,39);(9,11,14,22)(10,15,18,31)(12,24,26,20)(13,28,29,21)(16,32,23,30)(17,34,27,19)(25,37,40,39)(33,35,38,36);(2,5)(3,8)(6,7)(9,12,13,25,14,26,29,40)(10,16,17,33,18,23,27,38)(11,20,21,37,22,24,28,39)(15,30,19,35,31,32,34,36)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cf,d^b=dg,e^c=egh,f^c=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10446, #239899, #239918, … |
| maximal normal subgroup of | C2×#10446, #239899, #239918, … |
| maximal quotient of | #104944, #104945, #104959, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3434);