Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13304 |
| automorphism | Aut(G) = C27×#442 (order 8192) · Inn(G) = C2×D8 · Out(G) = C26×C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
223 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 223 conjugacy classes of subgroups — too many to draw. Only its 65 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 — 3 generators on 24 points:
(6,7)(9,10,13,17)(11,14,20,23)(12,21,19,16)(15,24,22,18);(1,2)(3,4)(5,6)(7,8);(2,4)(5,7,8,6)(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22)Polycyclic presentation:
2^8|a^2=h,c^2=d,e^2=gh,g^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,c^b=cf,e^c=eg,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13382, #498997, #498999, … |
| maximal normal subgroup of | C2×#13382, #498997, #498999, … |
| maximal quotient of | #144680, #144784, #144998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6370);