Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
391 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 125 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)(10,14)(11,17)(12,19)(15,22)(16,21)(20,23);(1,3,4,7)(2,5,6,8)(9,10,12,14,13,16,19,21)(11,15,18,20,17,23,24,22);(2,6)(5,8);(9,11,12,18,13,17,19,24)(10,15,14,20,16,23,21,22)Polycyclic presentation:
2^8|b^2=ef,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #265288, #265289, #265298, … |
| maximal normal subgroup of | #265288, #265289, #265298, … |
| maximal quotient of | #351270, #351271, #351276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26110);