Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7426 |
| automorphism | Aut(G) = #72231851 (order 1024) · Inn(G) = #1351 · Out(G) = C23 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1351 |
| commutator | C2×C22⋊C4 | C2×C4 |
| Frattini, cosocle | #379 | C22 |
| prime radical | C25 | D4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C24 ◁ #379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
166 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 166 conjugacy classes of subgroups — too many to draw. Only its 21 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) = 16 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2)(3,9)(4,12)(5,8)(6,10,15,13)(7,14)(11,16);(1,3)(2,6)(4,13,14,10)(5,11)(7,9,12,16)(8,15)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fg,b^a=bc,c^a=ce,e^a=eg,f^a=fh,c^b=cf,d^b=defg,f^b=fh,g^b=gh,d^c=dg,e^c=eh25 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7426, #150259, #150393, … |
| maximal normal subgroup of | C2×#7426, #150259, #150393, … |
| maximal quotient of | #93237, #93238, #93943, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 414);