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 | C22×D4 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1362 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)(4,6)(5,7)(9,10,12,14)(11,15,16,13)(17,18,23,27,22,25,29,32)(19,28,20,31,30,21,26,24);(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,23)(22,30)(24,27)(25,28)(26,29)(31,32);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,20)(18,24)(19,29)(21,32)(22,26)(23,30)(25,31)(27,28);(11,16)(13,15)(18,25)(19,30)(20,26)(27,32);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,21,29,31,22,28,23,24)(18,26,32,30,25,20,27,19)Polycyclic presentation:
2^9|a^2=h,c^2=fg,e^2=gh,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, 6314420);