Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 59 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) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,18,14,19,11,15)(12,17,20,24,21,16,22,23);(2,5)(6,8)(9,11,14,13)(10,15,19,18)(12,20,21,22)(16,23,17,24);(1,3,4,7)(2,6,5,8)(10,16)(12,21)(15,23)(17,19)(18,24)(20,22)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=d,e^2=h,f^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^c=eh,f^c=fgh,f^e=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94178, #94182, #94187, … |
| maximal normal subgroup of | #94178, #94182, #94187, … |
| maximal quotient of | #135166, #135169, #135207, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5849);