Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13331 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #459 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
216 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 55 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)(7,9)(8,12)(10,16)(11,15)(13,19)(14,17)(18,20);(6,12)(7,14)(8,9)(10,17)(13,16)(15,20)(18,19);(2,4)(5,7,8,16,17,20,19,15,11,13,18,14,10,12,9,6)Polycyclic presentation:
2^8|c^2=d,d^2=gh,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,c^b=cf,d^b=dg,e^b=egh,f^b=fg,g^b=gh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13445, #499182, #499183, … |
| maximal normal subgroup of | C2×#13445, #499182, #499183, … |
| maximal quotient of | #145365, #145367, #145368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6433);