Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17367 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1969 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 1969 conjugacy classes of subgroups — too many to draw. Only its 303 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 — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,12,9,10,15,17)(7,11,14,18,13,16,19,20);(6,10)(8,15)(11,16)(14,19);(2,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20);(7,13)(11,16)(14,19)(18,20)Polycyclic presentation:
2^8|a^2=g,g^2=h,b^a=be,c^a=cf,e^a=eh,g^b=gh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #128063, #128064, #128065, … |
| maximal normal subgroup of | #128063, #128064, #128065, … |
| maximal quotient of | #184748, #184750, #184753, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13580);