Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | Aut(G) order 196608 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C23 |
| commutator | C23 | C22×C8 |
| socle, prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
623 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(17,18,19,21,20,22,23,24);(2,4)(9,10,12,14)(11,15,16,13);(6,8)(9,11,12,16)(10,13,14,15)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,g^2=h,b^a=bd,c^a=ce,c^b=cf88 conjugacy classes — view the full character table.
| Fitting subgroup of | #11380⋊C3 |
| maximal subgroup of | #92886, #93485, #93522, … |
| maximal normal subgroup of | #92886, #93485, #93522, … |
| maximal quotient of | #121978, #121979, #121980, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4368);