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 | |
3843 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)(9,10,13,12)(11,15,16,14);(9,11,13,16)(10,14,12,15);(2,5)(6,8)(9,12,13,10)(11,15,16,14);(3,7)(6,8);(1,3)(2,6)(4,7)(5,8);(1,2)(3,6)(4,5)(7,8)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,b^a=bg,c^a=ch,f^c=fh,e^d=eh85 conjugacy classes — view the full character table.
| Fitting subgroup of | #62984⋊C3 |
| maximal subgroup of | #346520, #346528, #346536, … |
| maximal normal subgroup of | #346520, #346528, #346536, … |
| maximal quotient of | #7625774, #7625796, #7625799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55972);