Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12525 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,20)(21,30)(22,29)(23,26)(24,27)(25,32)(28,31);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(17,19,21,27,23,28,32,22)(18,24,25,20,26,29,30,31);(3,10)(6,15)(9,16)(11,14)(17,20,19,18,21,29,27,25,23,31,28,26,32,24,22,30)Polycyclic presentation:
2^8|a^2=fh,b^2=d,c^2=eg,d^2=f,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,c^b=cf46 conjugacy classes — view the full character table.
| maximal subgroup of | #123359, #123360, #123361, … |
| maximal normal subgroup of | #123359, #123360, #123361, … |
| maximal quotient of | #135786, #141712, #141719, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5522);