Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34732 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1773 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,14)(12,16)(13,15)(17,18,22,25,21,23,28,31)(19,20,24,29,26,27,30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(18,23)(20,27)(25,31)(29,32);(9,12,13,11)(10,14,15,16)(17,19)(18,20)(21,26)(22,24)(23,27)(25,29)(28,30)(31,32);(2,5)(6,8)(19,26)(20,27)(24,30)(29,32);(1,3,4,7)(2,6,5,8)(17,20,28,32,21,27,22,29)(18,24,31,19,23,30,25,26)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=dfi,e^b=ei,d^c=di,e^c=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6312797);