Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19279 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3038 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
382 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 382 conjugacy classes of subgroups — too many to draw. Only its 119 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 — 4 generators on 16 points:
(2,5)(3,7)(6,8)(10,14)(11,12)(15,16);(1,2,3,5,4,6,7,8);(2,6)(5,8)(9,10)(11,14)(12,16)(13,15);(9,11,13,12)(10,14,15,16)Polycyclic presentation:
2^8|b^2=e,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,c^b=cg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #271966, #271978, #271982, … |
| maximal normal subgroup of | #271966, #271978, #271982, … |
| maximal quotient of | #189116, #197801, #197802, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12279);