Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C4⋊C8 |
| prime radical | C23 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
151 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,20,15,19,31,36,16,21,30,18,13,32,12,28)(11,22,25,37,26,34,23,35,27,38,39,40,24,17,29,33);(2,5)(6,8)(9,11,13,24,16,27,15,26)(10,17,19,22,21,34,32,38)(12,29,30,39,31,23,14,25)(18,35,28,40,20,33,36,37);(1,3,4,7)(2,6,5,8)(9,12,13,30,16,31,15,14)(10,18,19,28,21,20,32,36)(11,23,24,25,27,29,26,39)(17,33,22,37,34,35,38,40)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ch64 conjugacy classes — view the full character table.
| maximal subgroup of | #93441, #93651, #93722, … |
| maximal normal subgroup of | #93441, #93651, #93722, … |
| maximal quotient of | #135635, #135639, #135645, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5872);