Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #385 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,9)(8,15,17,19)(10,13)(11,14)(12,16,20,18);(5,7,9,14,11,13,10,6)(8,15,16,20,17,19,18,12);(2,4)(5,8)(6,12)(7,15)(9,16)(10,18)(11,17)(13,19)(14,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dh,e^c=eh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10780, #353456, #354320, … |
| maximal normal subgroup of | C2×#10780, #353456, #354320, … |
| maximal quotient of | #105866, #105871, #105872, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3768);