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×B(4,𝔽2) | C2 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 446 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)(5,6,9,12)(7,13,15,20)(8,11,16,19)(10,17,18,14);(5,7)(6,10)(8,14)(9,15)(11,13)(12,18)(16,17)(19,20);(2,4)(8,16)(11,19)Polycyclic presentation:
2^8|a^2=h,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,g^b=gh,d^c=dg,f^d=fh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13344, #498885, #498886, … |
| maximal normal subgroup of | C2×#13344, #498885, #498886, … |
| maximal quotient of | #142217, #142218, #142265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6332);