Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | Aut(G) order 524288 |
| 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 | |
1051 conjugacy classes of subgroups; only the 391 normal subgroups are shown.
This group has 1051 conjugacy classes of subgroups — too many to draw. Only its 391 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(9,10,12,11)(13,14)(15,17)(16,19)(18,20);(5,6,7,8)(9,11,12,10)(14,15)(18,19);(1,2,3,4)(13,15,17,14)(16,19,20,18);(13,16)(14,18)(15,19)(17,20)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=f,b^a=be,c^a=cf,d^a=de,c^b=ce88 conjugacy classes — view the full character table.
| maximal subgroup of | #99153, #101052, #101056, … |
| maximal normal subgroup of | #99153, #101052, #101056, … |
| maximal quotient of | #153717, #153722, #153752, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6766);