Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | #2813 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1332 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)(3,8)(4,7)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18,24,22,23,27,21,32)(19,25,26,28,29,20,30,31);(2,6)(3,4)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,27)(21,30)(22,28)(23,29)(24,26)(31,32);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,24,31,23,25,21,28)(18,26,22,29,27,30,32,19);(1,4,5,3)(2,8,6,7)(9,11,12,16)(10,14,13,15)(17,21,23,24)(18,22,27,32)(19,26,29,30)(20,28,25,31);(2,6)(7,8)(17,22,24,27,23,32,21,18)(19,28,26,20,29,31,30,25)Polycyclic presentation:
2^9|a^2=h,c^2=fh,d^2=g,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dfi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338604);