Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3003 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
718 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 718 conjugacy classes of subgroups — too many to draw. Only its 139 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 — 4 generators on 20 points:
(6,10)(7,13)(8,17)(12,19)(15,16)(18,20);(1,2)(3,4)(5,6,8,10,9,12,17,19)(7,14,15,20,16,11,13,18);(2,4)(5,7)(6,11)(8,15)(9,16)(10,18)(12,14)(13,17)(19,20);(5,6,8,10,9,12,17,19)(7,11,15,18,16,14,13,20)Polycyclic presentation:
2^8|b^2=e,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,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #149945, #301949, #301950, … |
| maximal normal subgroup of | #149945, #301949, #301950, … |
| maximal quotient of | #262675, #262680, #262686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17266);