Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8608 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #380 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
243 conjugacy classes of subgroups; only the 83 normal subgroups are shown.
This group has 243 conjugacy classes of subgroups — too many to draw. Only its 83 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(2,5)(3,6)(7,8)(9,10,11,14,12,13,15,16);(1,2,6,8,4,7,3,5);(10,13)(11,15)Polycyclic presentation:
2^8|a^2=f,b^2=dg,d^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93011, #93030, #122815, … |
| maximal normal subgroup of | #93011, #93030, #122815, … |
| maximal quotient of | #98818, #98822, #98830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1604);