Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11499 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C24 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
328 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 328 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) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(1,3,4,7)(2,6,5,8)(10,15)(14,21)(16,19)(22,23)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=d,b^a=bd,c^a=ce,e^a=eg,g^a=gh,f^c=fg,f^e=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #133534, #133535, #133536, … |
| maximal normal subgroup of | #133534, #133535, #133536, … |
| maximal quotient of | #135162, #135176, #135205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5841);