Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3106 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
323 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 323 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,8)(3,4,7,5)(9,10,14,18)(11,15,20,24)(12,17,13,22)(16,21,23,19);(1,3,5,2,6,7,4,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,24)(18,23);(1,4,6,5)(2,7,8,3)(9,12,14,13)(10,17,18,22)(11,19,20,21)(15,16,24,23)Polycyclic presentation:
2^8|a^2=gh,b^2=e,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^b=dgh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #268643, #268644, #268647, … |
| maximal normal subgroup of | #268643, #268644, #268647, … |
| maximal quotient of | #189157, #197890, #197972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12477);