Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
1344 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,15)(14,16)(17,18,23,24,22,28,19,27)(20,30,31,32,26,25,29,21);(3,8)(5,7)(9,11,13,10)(12,15,16,14)(17,19,22,23)(18,24,28,27)(20,31,26,29)(21,25,32,30);(1,3,4,8)(2,5,6,7)(10,11)(14,15)(17,20)(18,25)(19,29)(21,24)(22,26)(23,31)(27,32)(28,30);(2,6)(5,7)(10,11)(14,15)(17,21,23,30,22,32,19,25)(18,26,24,29,28,20,27,31);(9,12,13,16)(10,14,11,15)(17,19,22,23)(18,27,28,24)(20,31,26,29)(21,30,32,25)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fg,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dfi,d^c=di,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6331666);