Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19186 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
325 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 123 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7)(3,5,8,6)(10,14)(11,18)(12,15)(17,22)(19,20)(21,24);(9,10,15,22,13,17,12,14)(11,16,18,24,19,23,20,21);(2,5)(3,8)(6,7)(9,11)(10,16)(12,20)(13,19)(14,21)(15,18)(17,23)(22,24);(11,19)(16,23)(18,20)(21,24)Polycyclic presentation:
2^8|a^2=gh,b^2=eg,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,d^c=dg,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #138582, #138589, #143338, … |
| maximal normal subgroup of | #138582, #138589, #143338, … |
| maximal quotient of | #189567, #197761, #202760, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12181);