Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2218 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×#102 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10,15,20,16,21,14,32)(11,22,26,34,27,39,25,37)(12,19,13,31,30,33,29,18)(17,23,35,40,36,28,38,24);(2,5)(6,7)(9,11,13,24,15,26,30,23,16,27,29,40,14,25,12,28)(10,17,33,39,32,38,31,34,21,36,19,22,20,35,18,37);(1,3,4,8)(2,6,5,7)(9,12,14,29,16,30,15,13)(10,18,20,19,21,31,32,33)(11,23,25,24,27,28,26,40)(17,34,35,39,36,37,38,22)Polycyclic presentation:
2^8|a^2=f,b^2=def,c^2=e,d^2=gh,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,c^b=ch52 conjugacy classes — view the full character table.
| maximal subgroup of | #131914, #131915, #131916, … |
| maximal normal subgroup of | #131914, #131915, #131916, … |
| maximal quotient of | #141885, #141886, #141910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6274);