Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #353 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C22×M16 | C22 |
| socle | C22 | #324 |
| prime radical | C24 | M16 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
171 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 55 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 — 2 generators on 20 points:
(5,6,8,13,10,11,16,18,9,14,17,20,7,12,15,19);(1,2,3,4)(6,11,14,12)(13,19,20,18)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=h,d^2=g,g^2=h,b^a=bc,c^a=cf,e^a=eh,f^a=fh,d^b=dfh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93897, #93908, #93912, … |
| maximal normal subgroup of | #93897, #93908, #93912, … |
| maximal quotient of | #92967, #92970, #92971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 181);