Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7349 |
| automorphism | Aut(G) = #72237748 (order 2048) · Inn(G) = #1317 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1317 |
| commutator | C24 | C2×C8 |
| Frattini, cosocle | #379 | C22 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,4,8)(3,6,10,13,9,14,11,16)(5,7,12,15);(1,3)(2,6,7,14)(4,11)(5,9)(8,13,15,16)(10,12)Polycyclic presentation:
2^8|a^2=d,b^2=g,d^2=f,b^a=bc,c^a=ce,e^a=eg,g^a=gh,c^b=ch,d^b=de,f^b=fh,d^c=dg,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94530, C2×#7349, #149552, … |
| maximal normal subgroup of | #94530, C2×#7349, #149552, … |
| maximal quotient of | #93684, #93708, #93909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 337);