Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | Aut(G) order 16384 |
| 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 | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
385 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 385 conjugacy classes of subgroups — too many to draw. Only its 127 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)(3,6)(4,5)(7,8)(9,10,16,18)(11,15,12,14)(13,20,22,19)(17,23,24,21);(2,5)(6,8)(9,11,14,10,16,12,15,18)(13,19,21,17,22,20,23,24);(1,3,4,7)(2,6,5,8)(9,12,14,18,16,11,15,10)(13,20,21,24,22,19,23,17);(9,13)(10,17)(11,19)(12,20)(14,21)(15,23)(16,22)(18,24)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gh,g^d=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #272417, #272420, #272424, … |
| maximal normal subgroup of | #272417, #272420, #272424, … |
| maximal quotient of | #350888, #350889, #350892, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26049);