Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C22×C22⋊C4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,15,13,11,16,14);(2,5)(6,8);(9,11)(10,13)(12,14)(15,16);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|a^2=g,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,f^c=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #105690, #105691, #105693, … |
| maximal normal subgroup of | #105690, #105691, #105693, … |
| maximal quotient of | #190582, #190583, #190584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13735);