Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #442 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C2×#509 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
222 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 222 conjugacy classes of subgroups — too many to draw. Only its 67 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)(6,12,16,15)(7,18,17,19)(8,13)(9,14)(10,20);(1,2,3,4)(5,6,13,19,20,7,8,15,11,16,14,18,10,17,9,12);(5,7,9,18,10,6,14,15,11,17,8,19,20,16,13,12)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=e,d^2=g,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gh,c^b=cg,d^b=dh,e^b=eh,f^b=fh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13249, #495240, #495241, … |
| maximal normal subgroup of | C2×#13249, #495240, #495241, … |
| maximal quotient of | #140749, #140751, #140784, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6237);