Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12028 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#106 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
153 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 153 conjugacy classes of subgroups — too many to draw. Only its 77 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:
(2,4)(5,6,8,13,10,14,9,19)(7,15,17,20,18,11,16,12);(6,11,14,15)(7,16)(8,9)(12,13,20,19)(17,18);(1,2,3,4)Polycyclic presentation:
2^8|a^2=f,b^2=fg,c^2=e,d^2=g,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,f^b=fh,g^b=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #136367, #136368, #136370, … |
| maximal normal subgroup of | #136367, #136368, #136370, … |
| maximal quotient of | #140388, #140389, #140403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6173);