Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13343 |
| automorphism | Aut(G) = C25×B(4,𝔽2) (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C23 |
| prime radical | C2×#453 | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
415 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 415 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(6,10)(8,15)(11,18)(12,16)(13,17)(19,20);(5,6)(7,11)(8,16)(9,12)(10,15)(13,20)(14,19)(17,18);(2,4)(5,7,9,14)(6,11,12,19)(8,17,15,13)(10,18,16,20)Polycyclic presentation:
2^8|c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fgh,c^b=cf,d^b=dh,g^b=gh,d^c=dg,f^d=fh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13347, #498922, #498923, … |
| maximal normal subgroup of | C2×#13347, #498922, #498923, … |
| maximal quotient of | #142217, #142222, #142264, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6335);