Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
795 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 795 conjugacy classes of subgroups — too many to draw. Only its 231 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:
(2,4)(6,7)(13,14,16,15);(1,2,3,4)(10,12);(5,6,8,7)(9,10)(11,12)(13,15,16,14);(1,2)(3,4)(5,7,8,6)Polycyclic presentation:
2^8|a^2=h,b^2=e,c^2=fh,d^2=f,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de64 conjugacy classes — view the full character table.
| Fitting subgroup of | #14562⋊C3 |
| maximal subgroup of | C2×#14562, #521024, #521047, … |
| maximal normal subgroup of | C2×#14562, #521024, #521047, … |
| maximal quotient of | #153971, #154005, #154047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 7550);