Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19279 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C22×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
331 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 331 conjugacy classes of subgroups — too many to draw. Only its 109 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 — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,19,24)(12,16,13,22)(15,20,23,21);(9,11,12,20,14,19,13,21)(10,15,22,24,17,23,16,18);(2,5)(3,8)(6,7)(11,19)(15,23)(18,24)(20,21);(2,6)(5,7)(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)Polycyclic presentation:
2^8|a^2=g,b^2=eg,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,c^b=cg,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #138451, #138458, #143096, … |
| maximal normal subgroup of | #138451, #138458, #143096, … |
| maximal quotient of | #189111, #197793, #202798, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12272);