Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
845 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 845 conjugacy classes of subgroups — too many to draw. Only its 187 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)(3,4)(9,10,11,14,12,13,15,16);(2,4)(5,6,7,8)(10,13)(11,15);(10,13)(11,15)Polycyclic presentation:
2^8|a^2=f,b^2=g,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^b=fh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #93477, #93483, #93521, … |
| maximal normal subgroup of | #93477, #93483, #93521, … |
| maximal quotient of | #105568, #105570, #105576, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3750);