Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11505 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
603 conjugacy classes of subgroups; only the 79 normal subgroups are shown.
This group has 603 conjugacy classes of subgroups — too many to draw. Only its 79 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6)(7,13,15,20)(8,16,18,11)(9,12)(10,14)(17,19);(5,7,9,15)(6,10,12,19)(8,17,18,14)(11,20,16,13);(2,4)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dh,f^b=fgh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94428, #94440, #133334, … |
| maximal normal subgroup of | #94428, #94440, #133334, … |
| maximal quotient of | #129216, #129248, #129338, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5682);