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 | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1484 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,10)(7,8,11,12)(13,14)(15,18)(16,19)(17,20);(5,7)(6,8)(9,11)(10,12)(13,15,16,20)(14,17,19,18);(1,2)(3,4)(6,10)(7,11)(13,15)(14,18)(16,20)(17,19);(5,7)(6,8)(9,11)(10,12)(13,14)(15,17)(16,19)(18,20);(5,8)(6,11)(7,10)(9,12)(13,15)(14,18)(16,20)(17,19)Polycyclic presentation:
2^8|a^2=fh,b^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 | #98194, #223010, #223038, … |
| maximal normal subgroup of | #98194, #223010, #223038, … |
| maximal quotient of | #518989, #518995, #519057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 29142);