Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7338 |
| automorphism | Aut(G) = #72237745 (order 2048) · Inn(G) = #1316 · Out(G) = C24 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1316 |
| commutator | C24 | C2×C8 |
| Frattini, cosocle | #379 | C22 |
| prime radical | C23×D4 | C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
245 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,8)(3,9,10,16,11,6,12,14)(5,7,13,15);(1,3)(2,6)(4,12)(5,11)(7,9)(8,14)(10,13)(15,16)Polycyclic presentation:
2^8|a^2=d,d^2=f,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=de,f^b=fh,d^c=dg,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94514, #94515, #94516, … |
| maximal normal subgroup of | #94514, #94515, #94516, … |
| maximal quotient of | #93191, #93193, #93202, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 326);