Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 786432 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
857 conjugacy classes of subgroups; only the 243 normal subgroups are shown.
This group has 857 conjugacy classes of subgroups — too many to draw. Only its 243 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) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(6,8)(9,10,12,14)(11,15,16,13)(17,18,22,20)(19,23,21,24);(5,6)(7,8)(17,19,22,21)(18,23,20,24);(1,2)(3,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,18)(19,23,21,24);(9,11,12,16)(10,13,14,15)(17,21,22,19)(18,23,20,24)Polycyclic presentation:
2^8|a^2=h,b^2=g,c^2=g,d^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| Fitting subgroup of | #15201⋊C3 |
| maximal subgroup of | #103243, #103259, #103275, … |
| maximal normal subgroup of | #103243, #103259, #103275, … |
| maximal quotient of | #154176, #154400, #154425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8189);