Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5212 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)(7,8)(9,10,12,11)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(10,11);(2,4)(9,11,12,10)(14,17)(18,20);(6,8)(10,11);(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|a^2=f,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72239620, #72262036, #72262072, … |
| maximal normal subgroup of | #72239620, #72262036, #72262072, … |
| maximal quotient of | #72239759, #72298425, #72298466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 518046);