Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| 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 | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 48 points:
(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,24,27,37,45,23,29,36,46,21,38,20,26)(19,30,33,44,32,25,31,41,34,43,47,40,39,42,35,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,34)(18,25,29,42)(20,35,37,31)(21,39,24,32)(22,33,36,47)(26,44,45,40)(27,43,38,30)(28,41,46,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20,21,36,23,37,24,22)(18,26,38,46,29,45,27,28)(19,31,39,33,34,35,32,47)(25,40,30,41,42,44,43,48)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=fg,d^2=h,e^2=g,g^2=h,b^a=bd,d^a=df,c^b=ch,d^b=dh,e^b=efh64 conjugacy classes — view the full character table.
| maximal subgroup of | #129676, #129677, #129678, … |
| maximal normal subgroup of | #129676, #129677, #129678, … |
| maximal quotient of | #124209, #124226, #124228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4486);