Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,18,21)(12,16,20,22)(17,23,19,24)(25,26,30,27,29,33,31,34)(28,36,39,37,38,35,40,32);(1,3,4,7)(2,5,6,8)(10,14)(15,21)(16,22)(23,24)(25,27,30,33,29,34,31,26)(28,37,39,35,38,32,40,36);(2,6)(5,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(25,28,29,38)(26,32,33,37)(27,35,34,36)(30,40,31,39);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(28,38)(32,37)(35,36)(39,40)Polycyclic presentation:
2^8|a^2=eg,b^2=f,c^2=e,d^2=eh,f^2=h,g^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #170271, #170273, #170303, … |
| maximal normal subgroup of | #170271, #170273, #170303, … |
| maximal quotient of | #380213, #380731, #381038, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23219);