Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 107 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:
(1,2,5,8)(3,4,7,6)(9,10,15,16,13,18,14,19)(11,20,22,23,17,12,21,24);(1,3,5,7)(2,6,8,4)(9,11)(10,12)(13,17)(14,21)(15,22)(16,24)(18,20)(19,23);(1,4,5,6)(2,7,8,3)(10,16)(11,21)(14,15)(17,22)(18,19)(23,24);(2,8)(3,7)(9,12)(10,17)(11,18)(13,20)(14,24)(15,23)(16,22)(19,21)Polycyclic presentation:
2^8|a^2=ef,b^2=eh,c^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #125722, #125724, #126217, … |
| maximal normal subgroup of | #125722, #125724, #126217, … |
| maximal quotient of | #344978, #344981, #344984, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24218);