Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of AΣL(1,𝔽16) |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #99 |
| commutator | C23 | C4×C8 |
| Frattini, cosocle | C22×C22⋊C4 | C22 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 63 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) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6,7,9,8,10,11,12)(13,14)(15,19,20,17)(16,18);(1,2,3,4)(14,17)(18,19)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=g,b^a=bc,c^a=cf,f^a=fh,d^b=df,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93892, #93901, #93909, … |
| maximal normal subgroup of | #93892, #93901, #93909, … |
| maximal quotient of | #92963, #92965, #92971, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 175);