Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×C22⋊C4 |
| commutator, prime radical | C23 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,27,23,24,30,31,22,26,29,32,19,28,20,25);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,22,23)(18,24,26,28)(20,21,30,29)(25,32,31,27);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,19,29,22,30,23,21)(18,25,28,32,26,31,24,27)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=fg,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=ch,e^b=ef64 conjugacy classes — view the full character table.
| maximal subgroup of | #129673, #129674, #129675, … |
| maximal normal subgroup of | #129673, #129674, #129675, … |
| maximal quotient of | #124209, #124226, #124228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4483);