Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996315 |
| 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 | #28251 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#125 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(10,14)(12,15)(17,18)(19,23)(20,27)(21,30)(22,25)(24,28)(26,31)(29,32);(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,12,16,15)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,27,31,32);(9,11)(10,15)(12,14)(13,16)(17,20,21,28,22,29,19,26)(18,24,23,27,25,31,30,32);(1,3,4,8)(2,6,5,7)(10,14)(11,16);(9,12,13,15)(10,11,14,16)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7106395);