Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,12,18,13,19,14,17)(11,16,20,15,21,23,22,24);(1,2,3,4)(9,11,12,20,13,21,14,22)(10,15,17,16,19,24,18,23);(6,8)(9,11,13,21)(10,16,19,23)(12,22,14,20)(15,18,24,17)Polycyclic presentation:
2^8|a^2=g,b^2=d,c^2=h,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^c=dh,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11128, #357812, #357814, … |
| maximal normal subgroup of | C2×#11128, #357812, #357814, … |
| maximal quotient of | #116986, #117004, #117006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4116);