Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8528 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
400 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2,6,8,4,3,5,7)(9,10,12,14,13,11,15,16);(5,6)(7,8)(9,11)(10,13)(12,14)(15,16);(1,3,4,2)(5,7,6,8)(12,15)(14,16)Polycyclic presentation:
2^8|a^2=f,c^2=g,d^2=h,e^2=g,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=egh,f^b=fgh,d^c=dgh,e^c=eg,f^c=fgh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94548, C2×#8532, #187896, … |
| maximal normal subgroup of | #94548, C2×#8532, #187896, … |
| maximal quotient of | #97046, #97050, #97262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1520);