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 | #2688 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1405 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,23,27,22,26,30,32)(19,24,25,31,20,21,28,29);(1,3,5,4)(2,6,7,8)(10,13)(11,15)(17,19,22,20)(18,24,26,21)(23,25,30,28)(27,31,32,29);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20,22,19)(18,24,26,21)(23,28,30,25)(27,31,32,29);(9,11,12,15)(10,14,13,16)(19,20)(21,24)(25,28)(29,31);(10,13)(14,16)(17,21,23,29,22,24,30,31)(18,25,27,20,26,28,32,19)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=fg,d^2=f,e^2=h,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, 6312713);