Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10362 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #393 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 208 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) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,18,20,24)(12,16)(13,22)(14,17)(15,21,23,19);(11,19,20,21)(15,18,23,24);(1,2,4,6)(3,8,7,5)(9,11)(10,15)(12,19)(13,21)(14,20)(16,24)(17,23)(18,22)Polycyclic presentation:
2^8|a^2=d,b^2=d,c^2=h,e^2=gh,f^2=g,b^a=bd,c^a=ce,e^a=eh,f^a=fg,c^b=cf,d^c=dg,e^c=egh,f^c=fg40 conjugacy classes — view the full character table.
| maximal subgroup of | #125958, #125959, #126253, … |
| maximal normal subgroup of | #125958, #125959, #126253, … |
| maximal quotient of | #104730, #104732, #104734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3351);