Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1665 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #389 |
| prime radical | C23 | C4⋊Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
149 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 75 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 — 3 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,16,14,17,12,21)(11,18,19,15,20,23,22,24);(9,11,13,19,14,20,12,22)(10,15,16,23,17,24,21,18);(1,2,3,5,4,6,7,8)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=e,d^2=g,e^2=h,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dg58 conjugacy classes — view the full character table.
| maximal subgroup of | #136486, #136487, #136572, … |
| maximal normal subgroup of | #136486, #136487, #136572, … |
| maximal quotient of | #101927, #101928, #101929, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2611);