Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C24 |
| commutator, socle | C22 | C24×C4 |
| Frattini, cosocle | C2×C4 | C25 |
| prime radical | #3482 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,23,21,25,19,24)(20,26,29,30,28,32,27,31);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,22)(18,23,25,24)(20,27,28,29)(26,30,32,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19,21,22)(18,24,25,23)(20,27,28,29)(26,31,32,30);(2,8)(6,9)(7,16)(14,15)(18,25)(23,24)(26,32)(30,31);(17,20,22,29,21,28,19,27)(18,26,23,30,25,32,24,31)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,b^a=bf,d^a=df,c^b=cfh100 conjugacy classes — view the full character table.
| Fitting subgroup of | #60331⋊C3 |
| maximal subgroup of | #129679, #210192, #210193, … |
| maximal normal subgroup of | #129679, #210192, #210193, … |
| maximal quotient of | #6370506, #6370507, #6370562, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53319);