Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1865 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(10,15)(11,12)(17,18,22,24)(19,20,21,23)(27,31)(29,32);(1,2,4,6)(3,5,8,7)(9,10,14,15)(11,13,12,16)(19,21)(20,23)(25,26,28,30)(27,29,31,32);(9,11,14,12)(10,16,15,13)(17,19,22,21)(18,23,24,20)(25,26)(27,29)(28,30)(31,32);(1,2)(3,5)(4,6)(7,8)(9,12,14,11)(10,16,15,13)(17,20,22,23)(18,19,24,21);(1,3)(2,5)(4,8)(6,7)(9,13,14,16)(10,11,15,12)(17,21,22,19)(18,23,24,20)(25,27)(26,29)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fhi,c^2=g,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6122779);