Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) = C27×#442 (order 8192) · Inn(G) = C2×D8 · Out(G) = C26×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(10,16)(11,20)(12,13)(14,22)(17,23)(18,24)(19,21);(5,7,8,6)(9,10,17,21,14,19,12,16,15,20,13,18,22,24,23,11);(2,4)(9,11,12,19,22,18,17,10,15,16,23,24,14,21,13,20)Polycyclic presentation:
2^8|a^2=f,b^2=dfgh,c^2=d,d^2=gh,e^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13150, #494907, #494908, … |
| maximal normal subgroup of | C2×#13150, #494907, #494908, … |
| maximal quotient of | #140177, #140202, #140234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6138);