Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | Aut(G) order 65536 |
| 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×#493 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
507 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 507 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) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,10,14,12,7,11,16)(8,17,18,15,19,20,9,13);(1,3,4,2)(5,7)(6,12)(8,15)(9,20)(10,14)(11,16)(13,19)(17,18);(5,8,10,18,12,19,11,9)(6,13,16,20,7,15,14,17);(5,9,11,19,12,18,10,8)(6,13,16,20,7,15,14,17)Polycyclic presentation:
2^8|a^2=g,b^2=egh,c^2=fh,d^2=eh,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,d^b=de,e^b=eh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #136488, #136512, #136537, … |
| maximal normal subgroup of | #136488, #136512, #136537, … |
| maximal quotient of | #231469, #231475, #231483, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16189);