Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7310 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1304 |
| commutator | C24 | C42 |
| Frattini, prime radical, cosocle | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
295 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 295 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,8,5,7,12,15)(3,6,9,13,10,14,11,16);(1,3)(2,6,7,14)(4,11,12,9)(5,10)(8,13)(15,16)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=h,b^a=bc,c^a=cf,e^a=eg,f^a=fh,g^a=gh,c^b=cg,d^b=df,f^b=fh,d^c=dh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94628, #94629, #94702, … |
| maximal normal subgroup of | #94628, #94629, #94702, … |
| maximal quotient of | #93100, #93105, #93107, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 299);