Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C22 | C22×C42 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
979 conjugacy classes of subgroups; only the 351 normal subgroups are shown.
This group has 979 conjugacy classes of subgroups — too many to draw. Only its 351 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10,13,11)(12,15,16,14)(18,21)(19,23);(10,11)(14,15)(17,18,20,21)(19,22,23,24);(1,2,3,4)(5,7,8,6)(9,11,13,10)(12,15,16,14)(17,18)(19,22)(20,21)(23,24);(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,23)(21,24)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=h,b^a=be,c^a=cf,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #99148, #101053, #101057, … |
| maximal normal subgroup of | #99148, #101053, #101057, … |
| maximal quotient of | #153717, #153719, #153855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6784);