Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479550 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17214 |
| 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 | |
660 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)(3,15,19,9)(4,20)(5,10)(6,14)(7,13)(8,12)(11,23)(16,31,24,29)(17,26,21,27)(18,28,25,32)(22,30);(1,3,6,18,8,19,7,25)(2,9,14,32,12,15,13,28)(4,21,23,16,22,17,5,24)(10,29,20,27,11,31,30,26);(1,4,6,23,8,22,7,5)(2,10,14,20,12,11,13,30)(3,16,18,17,19,24,25,21)(9,26,32,29,15,27,28,31);(1,5,6,22,8,23,7,4)(2,11,13,30,12,10,14,20)(3,17,18,16,19,21,25,24)(9,27,28,31,15,26,32,29)Polycyclic presentation:
2^9|a^2=fg,b^2=ef,c^2=ef,d^2=f,f^2=i,g^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=ei,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386746);