Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
203 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 203 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,14)(12,21)(13,16)(15,19)(17,20)(18,22)(23,24);(1,3,4,8)(2,5,6,7)(11,18)(14,22)(15,23)(19,24);(10,14)(11,19)(12,20)(15,17)(16,22)(18,24)(21,23)Polycyclic presentation:
2^8|a^2=d,b^2=d,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,e^b=eh,e^c=eg,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125884, #125886, #125927, … |
| maximal normal subgroup of | #125884, #125886, #125927, … |
| maximal quotient of | #144444, #144446, #144477, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6355);