Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499263 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13316 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
617 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,7,13,17,12,20,18,9,10,15,19,8,14,16,11);(1,2)(3,4)(6,10)(7,15)(12,14)(16,20);(2,4)(6,11)(7,16)(8,17)(10,18)(12,19)(13,14)(15,20);(6,12,10,14)(7,15)(11,19,18,13)(16,20)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fh,h^c=hi,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72250606, #72252782, #72252788, … |
| maximal normal subgroup of | #72250606, #72252782, #72252788, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406706);