Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17208 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,13,8,12,6,14)(3,15,18,27,20,9,19,31)(4,11,5,10,22,28,23,29)(16,32,17,25,24,26,21,30);(1,3,6,18,8,20,7,19)(2,9,14,31,12,15,13,27)(4,21,23,16,22,17,5,24)(10,26,11,30,29,32,28,25);(1,4,6,23,8,22,7,5)(2,10,14,11,12,29,13,28)(3,16,18,17,20,24,19,21)(9,25,31,26,15,30,27,32);(1,5,6,22,8,23,7,4)(2,11,13,29,12,28,14,10)(3,17,18,16,20,21,19,24)(9,26,27,30,15,32,31,25)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ef,d^2=f,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386660);