Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D8 |
| commutator | C8 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle, prime radical | C23 | #107 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 28 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(14,19)(15,23)(16,25)(17,24)(20,21)(22,28)(26,27);(2,4)(13,14,16,20,17,21,15,19,18,22,23,27,24,26,25,28)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=f,f^2=h,b^a=bc,c^a=cf,e^a=efh,f^a=fh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93915, #93916, #93931, … |
| maximal normal subgroup of | #93915, #93916, #93931, … |
| maximal quotient of | #92911, #92915, #92923, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 142);