Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C27×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(5,6,8,7)(14,17)(18,20);(9,10,11,12)(13,14)(15,18)(16,17)(19,20);(2,4)(5,6,8,7);(5,7,8,6)(14,17)(18,20);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,d^b=df200 conjugacy classes — view the full character table.
| maximal subgroup of | #72280376, #72280386, #72280449, … |
| maximal normal subgroup of | #72280376, #72280386, #72280449, … |
| maximal quotient of | C2×#513485 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420681);