Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
333 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 139 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:
(1,2)(3,7)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(18,21)(23,24);(2,5)(6,7)(17,18,22,23,19,21,20,24);(9,11,12,16)(10,13,14,15)(18,23)(20,22)(21,24);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=f,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #305897, #305898, #305905, … |
| maximal normal subgroup of | #305897, #305898, #305905, … |
| maximal quotient of | #362175, #363809, #367521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21787);