Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(10,14)(15,16);(9,10)(11,15)(12,16)(13,14);(2,5)(6,8);(3,7)(6,8)(9,11,13,12)(10,15,14,16);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,16,14,15);(1,2)(3,6)(4,5)(7,8)Polycyclic presentation:
2^8|d^2=g,e^2=g,b^a=bg,c^a=ch,f^c=fh,e^d=eh85 conjugacy classes — view the full character table.
| Fitting subgroup of | #62985⋊C3 |
| maximal subgroup of | #347232, #347256, #347280, … |
| maximal normal subgroup of | #347232, #347256, #347280, … |
| maximal quotient of | #7625775, #7625776, #7625800, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55973);