Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10,13,16)(11,12,15,14)(17,18,22,21)(19,20,24,23)(25,26)(27,30)(28,32)(29,31);(9,11,13,15)(10,14,16,12)(17,19)(18,23)(20,21)(22,24)(25,27,29,32)(26,30,31,28);(11,15)(12,14)(18,21)(19,24)(25,27)(26,28)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,28)(29,32)(30,31);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,16,11)(17,21,22,18)(19,20,24,23)(25,28)(26,27)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=f,b^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4653871);