Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #36614 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2421 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,15,16,13)(12,14)(17,18)(19,23)(20,28,29,31)(21,25)(22,26)(24,27,32,30);(1,3)(2,5)(4,8)(6,7)(11,16)(13,15)(17,19,22,21)(18,23,26,25)(20,27,29,30)(24,28,32,31);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,24)(19,27)(21,30)(22,29)(23,28)(25,31)(26,32);(2,6)(5,7)(17,19,22,21)(18,25,26,23)(20,27,29,30)(24,31,32,28)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121571);