Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #365 |
| automorphism | Aut(G) order 2048 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #106 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C2 | #1279 |
| prime radical | C23 | #104 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,11,7,13,23,28,8,14,22,10,6,12,4,21)(3,15,17,26,19,31,16,27,20,29,24,25,18,30,32,9);(1,3)(2,9,12,25,14,27,13,26)(4,16)(5,24)(6,18)(7,19)(8,20)(10,15,28,30,11,29,21,31)(17,22)(23,32)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=fg,f^2=h,g^2=h,b^a=bc,c^a=cf,e^a=ef,f^a=fh,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7180, #126851, #126872, … |
| maximal normal subgroup of | C2×#7180, #126851, #126872, … |
| maximal quotient of | #92933, #92934, #92935, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 168);