Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10260 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
311 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(14,19)(15,20)(16,17);(5,6)(7,9)(8,11)(10,12)(13,14)(15,16)(17,19)(18,20);(2,4)(6,9)(7,12)(10,11)(13,15,16,14,18,19,17,20)Polycyclic presentation:
2^8|c^2=d,d^2=g,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,c^b=cf,d^b=dg,e^b=eh,f^b=fh,e^c=egh,f^c=fgh46 conjugacy classes — view the full character table.
| maximal subgroup of | #125693, #125694, #125695, … |
| maximal normal subgroup of | #125693, #125694, #125695, … |
| maximal quotient of | #104857, #104858, #105073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3524);