Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24283 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3036 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 435 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(6,11)(7,14)(8,18)(9,12)(16,17)(19,20);(1,2,3,4)(5,6,12,18,10,8,9,11)(7,15,14,19,17,13,16,20);(2,4)(5,7)(6,13)(8,15)(9,16)(10,17)(11,19)(12,14)(18,20);(5,8,9,18,10,6,12,11)(7,15,16,20,17,13,14,19)Polycyclic presentation:
2^8|b^2=eg,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #332328, #332331, #332338, … |
| maximal normal subgroup of | #332328, #332331, #332338, … |
| maximal quotient of | #317966, #317967, #317970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21230);