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 | #52139 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1732 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)(3,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,27,22,26,28,31)(19,21,20,30,29,32,25,24);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19,22,29)(18,21,26,32)(20,28,25,23)(24,27,30,31);(10,14)(13,16)(17,20)(18,24)(19,28)(21,27)(22,25)(23,29)(26,30)(31,32);(2,5)(6,8)(19,29)(20,25)(21,32)(24,30);(1,3,4,7)(2,6,5,8)(17,21,23,30,22,32,28,24)(18,25,27,19,26,20,31,29)Polycyclic presentation:
2^9|a^2=h,b^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,e^a=efi,d^b=dfi,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, 6309405);