Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11677 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
185 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 185 conjugacy classes of subgroups — too many to draw. Only its 87 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 — 3 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(10,16,19,23)(11,21)(12,22)(13,14)(17,20,24,18);(2,5)(6,7)(9,10,11,17,13,18,21,16,15,19,22,24,14,20,12,23);(1,3,4,8)(2,6,5,7)(9,11,13,21,15,22,14,12)(10,17,18,16,19,24,20,23)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,d^2=g,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #131931, #131932, #131961, … |
| maximal normal subgroup of | #131931, #131932, #131961, … |
| maximal quotient of | #141887, #141889, #141914, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6275);