Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8304 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
356 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 356 conjugacy classes of subgroups — too many to draw. Only its 81 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,9,11,8,7,10,12)(13,14,17,20,18,19,16,15);(5,7,9,12,8,6,10,11)(13,15,17,19,18,20,16,14);(2,4)(6,7)(11,12)(13,16,18,17)(14,15,19,20)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dg,c^b=ch,d^b=dg,f^b=fg52 conjugacy classes — view the full character table.
| maximal subgroup of | #97798, #97800, #97802, … |
| maximal normal subgroup of | #97798, #97800, #97802, … |
| maximal quotient of | #99756, #99757, #99768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2072);