Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#120 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6,10,9)(7,12,8,11)(14,18)(19,20);(2,4)(5,7,10,8)(6,11,9,12)(13,14,17,18)(15,19,16,20);(1,2)(3,4)(5,8,10,7)(6,12,9,11);(2,4)(5,6,10,9)(7,11,8,12)(13,15,17,16)(14,19,18,20);(1,2)(3,4)(5,9,10,6)(7,12,8,11)(13,16,17,15)(14,20,18,19)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^b=eh,d^c=dh,e^d=eh70 conjugacy classes — view the full character table.
| maximal subgroup of | #98211, #219863, #219883, … |
| maximal normal subgroup of | #98211, #219863, #219883, … |
| maximal quotient of | #519101, #519111, #519122, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29121);