Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431723 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62853 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,3)(5,6)(7,12)(8,11)(9,13)(10,14)(15,19)(16,18)(17,20);(1,2)(3,4)(5,6)(7,12)(8,13)(9,11)(10,14)(15,19)(16,20)(17,18);(1,3,4,2)(6,11)(8,9)(12,18)(13,14)(16,17)(19,20);(5,7)(6,12)(8,16)(9,17)(10,15)(11,18)(13,20)(14,19);(7,15)(12,19)(16,17)(18,20)Polycyclic presentation:
2^9|c^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,h^b=hi,g^c=gi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339086);