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×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 390 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,10,8)(7,13,16,20)(9,17,19,12)(11,18,14,15);(5,7)(6,11)(8,14)(9,18)(10,16)(12,20)(13,17)(15,19);(2,4)(5,8,10,6)(7,14,16,11)(9,12,19,17)(13,15,20,18)Polycyclic presentation:
2^8|a^2=h,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×#13349, #498922, #498924, … |
| maximal normal subgroup of | C2×#13349, #498922, #498924, … |
| maximal quotient of | #142219, #142223, #142266, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6337);