Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7348 |
| automorphism | Aut(G) = #72284297 (order 2048) · Inn(G) = #1317 · Out(G) = C2×D4 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1317 |
| commutator | C22×C4 | C2×C8 |
| Frattini, cosocle | #407 | C22 |
| prime radical | C24 | M16 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #407 ◁ G | |
| LECS | G ▷ #407 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
89 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,10,6,9,19,25,7,11,18,28,4,16,17,27)(3,12,14,29,13,8,21,22,15,24,30,32,20,23,31,26);(1,3)(2,8,11,23)(4,13)(5,14)(6,20)(7,15)(9,12,16,24)(10,26,28,22)(17,21)(18,30)(19,31)(25,32,27,29)Polycyclic presentation:
2^8|a^2=d,b^2=g,c^2=h,d^2=f,f^2=h,b^a=bc,c^a=ce,e^a=eg,g^a=gh,d^b=deh,f^b=fh,d^c=dg,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94528, #94529, #94552, … |
| maximal normal subgroup of | #94528, #94529, #94552, … |
| maximal quotient of | #93683, #93708, #93910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 338);