Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1737 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(10,13)(11,15)(18,21)(19,23)(25,26)(27,30)(28,32)(29,31);(1,2,5,3)(4,7,8,6)(9,10,12,13)(11,14,15,16)(25,27,29,32)(26,30,31,28);(2,3)(6,7)(11,15)(14,16)(17,18)(19,24)(20,21)(22,23)(25,28)(26,27)(29,30)(31,32);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,14,13,16)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,32)(29,31);(1,4,5,8)(2,6,3,7)(9,10)(11,14)(12,13)(15,16)Polycyclic presentation:
2^9|b^2=gi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4862862);