Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10012 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| socle | C22 | #399 |
| prime radical | C23 | C22⋊Q8 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,6,8,5,3,4,7)(9,10,14,22,13,17,12,16)(11,18,19,15,21,24,20,23);(1,3,5,2)(4,7,6,8)(9,11,12,19,13,21,14,20)(10,15,16,24,17,23,22,18);(1,2,5,3)(4,8,6,7)(9,12,13,14)(10,16,17,22)(11,19,21,20)(15,18,23,24)Polycyclic presentation:
2^8|a^2=dg,b^2=de,c^2=g,d^2=g,e^2=h,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,d^b=dg,e^b=eg,d^c=dh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #93244, #93248, #126443, … |
| maximal normal subgroup of | #93244, #93248, #126443, … |
| maximal quotient of | #105233, #105238, #105239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3651);