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 | #537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 325 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)(3,4)(5,7)(6,8)(9,10,15,16,14,19,13,18)(11,12,21,23,17,24,20,22);(1,3)(2,6)(4,7)(5,8)(9,11,14,17)(10,12,19,24)(13,20,15,21)(16,23,18,22);(1,4,5,6)(2,3,7,8)(10,16)(11,20)(13,15)(17,21)(18,19)(22,23);(2,7)(3,8)(9,12)(10,17)(11,19)(13,23)(14,24)(15,22)(16,21)(18,20)Polycyclic presentation:
2^8|a^2=fh,b^2=h,c^2=e,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 | #151245, #151247, #151417, … |
| maximal normal subgroup of | #151245, #151247, #151417, … |
| maximal quotient of | #343427, #343429, #343433, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24100);