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 | |
4027 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 16 points:
(1,2)(3,6)(4,5)(7,8)(10,13)(14,16);(9,10)(11,14)(12,13)(15,16);(2,5)(6,8)(9,11,12,15)(10,14,13,16);(3,7)(6,8);(1,3)(2,6)(4,7)(5,8);(1,2)(3,6)(4,5)(7,8)Polycyclic presentation:
2^8|c^2=g,b^a=bg,c^a=ch,f^c=fh,e^d=eh85 conjugacy classes — view the full character table.
| maximal subgroup of | #346519, #346527, #346535, … |
| maximal normal subgroup of | #346519, #346527, #346535, … |
| maximal quotient of | #7625773, #7625774, #7625795, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55971);