Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 50331648 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C24×#41 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
18790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,7,8)(9,10,11,12)(15,19)(18,20);(5,6)(7,8)(14,17)(18,20);(9,10)(11,12)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|a^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
| maximal subgroup of | #72337230, #72337232, #72337253, … |
| maximal normal subgroup of | #72337230, #72337232, #72337253, … |
| maximal quotient of | #72337238, #72337388, C2×#4308083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215279);