Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1599 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)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,26,29,24)(20,21,25,30)(27,28,32,31);(1,3)(2,5)(4,8)(6,7)(17,19,22,29)(18,24,23,26)(20,28,25,31)(21,27,30,32);(9,11,12,16)(10,13,14,15)(17,20)(18,21)(19,27)(22,25)(23,30)(24,31)(26,28)(29,32);(2,6)(5,7);(3,8)(5,7)(17,21,22,30)(18,25,23,20)(19,28,29,31)(24,27,26,32)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6595257);