Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2196 |
| automorphism | Aut(G) = C25×B(4,𝔽2) (order 2048) · Inn(G) = B(4,𝔽2) · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator | C2×D4 | C22×C4 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C2×H(5,𝔽2) | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(2,4)(5,6)(7,13,14,19)(8,11,16,18)(9,12)(10,15)(17,20);(1,2,3,4)(11,18)(13,19);(5,7,9,14)(6,10,12,17)(8,15,16,20)(11,13,18,19)Polycyclic presentation:
2^8|a^2=f,b^2=df,c^2=h,b^a=bd,c^a=ce,e^a=eg,g^a=gh,e^b=egh,d^c=dg,f^c=fg,e^d=eh,f^e=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#13042, #494571, #494576, … |
| maximal normal subgroup of | C2×#13042, #494571, #494576, … |
| maximal quotient of | #136818, #136846, #137066, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6030);