Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C22×#120 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1254 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 1254 conjugacy classes of subgroups — too many to draw. Only its 79 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10);(6,9)(10,11);(2,4)Polycyclic presentation:
2^8|a^2=f,b^a=bd,c^a=ce,d^a=dg,g^a=gh,f^b=fg,f^d=fh40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5675);