Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | C2×#525 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16,12,14,11,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,32)(29,31);(1,3)(2,5)(4,7)(6,8)(10,14)(15,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,31)(25,26)(30,32);(2,6)(3,7)(9,11,12,13)(10,15,14,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32);(2,6)(5,8)(19,28)(23,26)(27,29)(31,32);(9,11,12,13)(10,15,14,16)(17,21,22,20)(18,24,25,30)(19,29,28,27)(23,31,26,32)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dfi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315559);