Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,15,16,13);(1,3,5,4)(2,6,8,7)(11,16)(13,15)(17,18,22,25)(19,23,26,30)(20,27,29,32)(21,28,31,24);(9,11,12,16)(10,13,14,15)(17,19,22,26)(18,23,25,30)(20,28,29,24)(21,27,31,32);(10,14)(13,15)(17,20,21,30,22,29,31,23)(18,19,24,27,25,26,28,32);(1,4,5,3)(2,7,8,6)(19,26)(20,29)(23,30)(27,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=gi,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7522590);