Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10078 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #120 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
191 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 191 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,5,7)(3,8,4,6)(9,10,12,15)(11,13,16,14)(19,24)(21,23);(1,3,5,4)(2,6,7,8)(11,16)(13,14)(17,18)(19,21)(20,22)(23,24);(1,4,5,3)(2,6,7,8)(10,13)(11,16)(14,15)(18,21)(19,24)(22,23)Polycyclic presentation:
2^8|a^2=dg,b^2=d,c^2=d,e^2=g,f^2=gh,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fgh,e^c=eg,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94728, C2×#10585, #241234, … |
| maximal normal subgroup of | #94728, C2×#10585, #241234, … |
| maximal quotient of | #105173, #105174, #105176, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3573);