Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10637 |
| automorphism | Aut(G) = C23×#7737411 (order 4096) · Inn(G) = #397 · Out(G) = C26 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #397 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
165 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 51 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:
(1,2,5,6,4,8,3,7)(9,10,14,15,13,18,12,17)(11,19,21,16,20,23,22,24);(10,15)(11,20)(12,14)(16,19)(17,18)(23,24);(2,6)(3,5)(7,8)(9,11,12,21,13,20,14,22)(10,16,17,23,18,24,15,19)Polycyclic presentation:
2^8|a^2=eh,c^2=d,d^2=g,e^2=h,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,e^c=eh,f^c=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10641, #243017, #243024, … |
| maximal normal subgroup of | C2×#10641, #243017, #243024, … |
| maximal quotient of | #105215, #105217, #105225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3629);