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) = C23 |
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 | |
320 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,17,19,24)(12,15,20,23)(14,21,22,18);(1,3,4,8)(2,5,6,7)(12,20)(15,23);(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,21)(23,24)Polycyclic presentation:
2^8|a^2=dh,b^2=d,b^a=bd,c^a=ce,f^a=fgh,c^b=cf,e^b=eg,g^c=gh,f^e=fh31 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13376, #498980, #498981, … |
| maximal normal subgroup of | C2×#13376, #498980, #498981, … |
| maximal quotient of | #144433, #144434, #144437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6364);