Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C2×D4 |
| commutator | C4 | C43 |
| Frattini, cosocle | C23×C4 | C23 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
571 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 571 conjugacy classes of subgroups — too many to draw. Only its 271 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,7,8)(14,18)(15,16)(19,20);(2,4)(13,14,16,20,17,19,15,18);(9,10,11,12)(14,19)(18,20)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=g,d^2=h,f^2=h,b^a=bd,d^a=dh,f^a=fh,c^b=ch88 conjugacy classes — view the full character table.
| maximal subgroup of | #107436, #107437, #107704, … |
| maximal normal subgroup of | #107436, #107437, #107704, … |
| maximal quotient of | #105272, #105274, #105276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3691);