Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12019 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#106 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C23 | C2×SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,15,17,27)(8,19)(9,12)(10,23)(11,16,24,29)(13,21)(14,22)(18,32,31,28)(20,33)(25,36,35,30)(26,34);(5,7,8,16,10,18,21,30,9,17,20,29,22,31,34,36)(6,11,13,15,14,25,19,28,12,24,26,27,23,35,33,32);(1,2,3,4)(6,12)(11,24)(13,26)(14,23)(15,27)(19,33)(25,35)(28,32)Polycyclic presentation:
2^8|a^2=f,b^2=d,c^2=eh,d^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #134945, #134946, #134949, … |
| maximal normal subgroup of | #134945, #134946, #134949, … |
| maximal quotient of | #135716, #135717, #135742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5892);