Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,23,26,22,21,29,30)(19,24,27,31,20,25,28,32);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,20)(18,24,21,25)(23,27,29,28)(26,31,30,32);(11,15)(13,16)(17,20,22,19)(18,25,21,24)(23,27,29,28)(26,31,30,32);(1,3,4,7)(2,6,5,8)(17,21)(18,22)(19,25)(20,24)(23,26)(27,31)(28,32)(29,30)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=i,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,h^c=hi,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358202);