Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7304 |
| automorphism | Aut(G) = #72285069 (order 4096) · Inn(G) = #342 · Out(G) = #379 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #342 |
| commutator | C24 | C42 |
| Frattini, cosocle | C22×C22⋊C4 | C22 |
| prime radical | C25 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
241 conjugacy classes of subgroups; only the 39 normal subgroups are shown.
This group has 241 conjugacy classes of subgroups — too many to draw. Only its 39 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) = 32 |
A minimal faithful permutation representation — 2 generators on 16 points:
(2,5)(6,7)(9,10,12,13)(11,14)(15,16);(1,2)(3,7,8,5)(4,6)(9,11,13,16,12,15,10,14)Polycyclic presentation:
2^8|a^2=d,b^2=e,e^2=f,b^a=bc,c^a=cf,e^a=eg,c^b=cg,d^b=df,g^b=gh,e^c=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7324, #128156, #128157, … |
| maximal normal subgroup of | C2×#7324, #128156, #128157, … |
| maximal quotient of | #93464, #93496, #93557, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 312);