Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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)(17,18,22,26,21,24,30,29)(19,27,28,20,25,23,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,19,21,25)(18,23,24,27)(20,26,31,29)(22,28,30,32);(10,14)(13,16)(17,19)(18,23)(20,29)(21,25)(22,28)(24,27)(26,31)(30,32);(2,5)(6,8)(18,24)(19,25)(26,29)(28,32);(1,3,4,7)(2,6,5,8)(17,20,30,27,21,31,22,23)(18,25,29,28,24,19,26,32)Polycyclic presentation:
2^9|a^2=h,b^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,c^b=ci,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6309564);