Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7348 |
| 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 | C22×C4 | C2×C8 |
| Frattini, cosocle | #381 | C22 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #381 ◁ G | |
| LECS | G ▷ #381 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,9)(3,11,13,27,14,28,30,32)(4,15,16,25)(6,10,18,26)(7,19,20,29,21,31,24,12)(8,22,23,17);(1,3)(2,7,10,21)(4,12)(5,13)(6,14)(8,11,15,28)(9,24,26,20)(16,19)(17,29)(18,30)(22,31)(23,32,25,27)Polycyclic presentation:
2^8|a^2=d,b^2=g,c^2=h,d^2=f,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 | #94526, #94527, #94550, … |
| maximal normal subgroup of | #94526, #94527, #94550, … |
| maximal quotient of | #93683, #93707, #93909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 336);