Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #399 |
| prime radical | C22×D4 | Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
186 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 186 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,7,8,5)(9,10,12,15,13,16,18,22)(11,17,19,23,20,24,21,14);(2,5)(3,8)(6,7)(9,11,18,19,13,20,12,21)(10,14,22,17,16,23,15,24);(2,5)(3,8)(6,7)(14,23)(17,24)Polycyclic presentation:
2^8|a^2=d,b^2=deg,d^2=g,e^2=h,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93241, #93245, #126283, … |
| maximal normal subgroup of | #93241, #93245, #126283, … |
| maximal quotient of | #105229, #105230, #105231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3648);