Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2541 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
606 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 606 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3,4)(10,14)(11,15)(12,16);(1,2)(3,4)(5,6)(7,8)(10,11)(14,15);(6,8)(9,10,13,11)(12,15,16,14);(2,4)(9,11,13,10)(12,14,16,15)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=def,f^a=fh,c^b=cg,d^b=deh,f^c=fh,f^d=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | #142497, #142508, #143020, … |
| maximal normal subgroup of | #142497, #142508, #143020, … |
| maximal quotient of | #332963, #332964, #332965, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23504);