Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D8 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #335 |
| prime radical | C23 | #108 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2)(3,4)(5,6,8,15,11,16,24,30,12,17,25,31,10,26,9,14)(7,18,19,13,22,27,34,28,23,33,35,29,21,36,20,32);(2,4)(5,7,9,20,10,21,25,35,12,23,24,34,11,22,8,19)(6,13,15,27,16,28,30,33,17,29,31,36,26,32,14,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=f,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=efh,f^a=fh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93917, #93919, #93920, … |
| maximal normal subgroup of | #93917, #93919, #93920, … |
| maximal quotient of | #92911, #92912, #92917, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 148);