Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1509 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(18,23)(20,21)(24,30)(25,29)(27,28)(31,32);(1,3)(2,5)(4,8)(6,7)(9,10,13,16)(11,15,12,14)(19,26)(24,31)(27,28)(30,32);(3,8)(5,7)(9,11,13,12)(10,14,16,15)(17,18)(19,24)(20,23)(21,29)(22,25)(26,31)(27,32)(28,30);(2,6)(5,7)(17,19,22,26)(18,24,25,31)(20,28,21,27)(23,30,29,32);(9,12,13,11)(10,15,16,14)(17,20,22,21)(18,23,25,29)(19,27,26,28)(24,32,31,30)Polycyclic presentation:
2^9|b^2=fh,c^2=fh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6759760);