Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8760 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #381 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
211 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,19,13,20,14,18)(11,16,21,17,12,22,23,24);(2,5)(6,7)(9,11,13,12)(10,16,20,22)(14,21,15,23)(17,19,24,18);(2,6)(4,8)(5,7)(9,12,13,11)(10,17)(14,23,15,21)(16,19)(18,22)(20,24)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=dh,e^2=gh,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,e^c=eh,f^c=fg43 conjugacy classes — view the full character table.
| maximal subgroup of | #130892, #130893, #130894, … |
| maximal normal subgroup of | #130892, #130893, #130894, … |
| maximal quotient of | #99169, #99171, #99185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1755);