Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#472 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 446 conjugacy classes of subgroups — too many to draw. Only its 167 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:
(1,2,4,3)(5,6,9,12,10,7,14,13)(8,15,17,16,18,11,19,20);(1,3,4,2)(5,7)(6,10)(8,16)(9,12)(11,19)(13,14)(15,17)(18,20);(5,6)(7,10)(8,11)(9,12)(13,14)(15,18)(16,19)(17,20);(5,8,9,17,10,18,14,19)(6,11,12,20,7,15,13,16)Polycyclic presentation:
2^8|a^2=g,b^2=egh,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,e^b=eh,g^b=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #498126, #498379, #504519, … |
| maximal normal subgroup of | #498126, #498379, #504519, … |
| maximal quotient of | #229186, #229192, #229199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16036);