Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | Aut(G) = C2×#72311718 (order 4096) · Inn(G) = #447 · Out(G) = C23×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #447 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| prime radical | C24 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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:
(1,2,4,3)(6,10)(7,13)(8,16)(11,18)(12,19)(14,17)(15,20);(1,3,4,2)(5,6)(7,14)(8,17)(9,11)(10,13)(12,16)(15,19)(18,20);(7,15)(10,18)(13,20)(14,19)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gh,d^b=dg,e^b=eh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #135990, #135991, #135993, … |
| maximal normal subgroup of | #135990, #135991, #135993, … |
| maximal quotient of | #135816, #135842, #135982, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5911);