Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33348 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#99 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C2×D4 | C24 |
| prime radical | #3484 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,7)(4,11,12,6)(5,8)(9,14)(10,16,15,13);(2,6)(7,13)(8,11)(14,16);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15);(3,9)(7,14)(10,15)(13,16)Polycyclic presentation:
2^8|a^2=f,b^a=be,e^a=eg,g^a=gh,f^b=fg,d^c=dh,f^e=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #493474, #493476, #493477, … |
| maximal normal subgroup of | #493474, #493476, #493477, … |
| maximal quotient of | #480010, #480050, #480097, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26336);