Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478819 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17170 |
| 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:
(1,2,7,8,6,12,5,11)(3,9,20,24,15,26,14,22)(4,16,17,27,19,23,21,10)(13,29,18,25,31,28,30,32);(2,8)(4,17)(5,7)(9,24)(10,27)(11,12)(13,30)(14,20)(18,31)(19,21)(22,26)(28,29);(1,3,5,20,6,15,7,14)(2,9,8,22,12,26,11,24)(4,18,17,13,19,30,21,31)(10,28,23,32,27,29,16,25);(1,4,5,17,6,19,7,21)(2,10,11,16,12,27,8,23)(3,13,14,18,15,31,20,30)(9,25,22,28,26,32,24,29)Polycyclic presentation:
2^9|a^2=efg,c^2=ef,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386015);