Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
383 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 383 conjugacy classes of subgroups — too many to draw. Only its 159 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,24)(19,20,21,23);(2,5)(6,8)(17,18,21,20,22,24,19,23);(9,11,12,16)(10,13,14,15)(17,19,22,21)(18,23,24,20);(1,3,4,7)(2,6,5,8)(17,20,21,24,22,23,19,18)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #104443, #104447, #104459, … |
| maximal normal subgroup of | #104443, #104447, #104459, … |
| maximal quotient of | #392485, #393406, #393415, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26281);