Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8626 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #380 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #117 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
168 conjugacy classes of subgroups; only the 69 normal subgroups are shown.
This group has 168 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:
(1,2,6,9,7,4,5,15)(3,10,13,11,14,16,12,8)(18,22)(19,26)(20,23)(25,30)(27,28)(29,31);(1,3,5,12,7,14,6,13)(2,8,9,10,4,11,15,16)(17,18,23,30,21,25,20,22)(19,24,26,31,28,32,27,29);(1,4)(2,7)(3,11)(5,15)(6,9)(8,14)(10,13)(12,16)(17,19,21,28)(18,24,25,32)(20,27,23,26)(22,29,30,31)Polycyclic presentation:
2^8|a^2=f,b^2=dh,c^2=h,d^2=g,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,e^c=eg,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #130470, #130471, #131072, … |
| maximal normal subgroup of | #130470, #130471, #131072, … |
| maximal quotient of | #98841, #98843, #98849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1629);