Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1582 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,8)(4,6)(11,15)(14,16)(17,18,22,24)(19,21,23,20)(25,26,29,27)(28,31,32,30);(1,2,5,7)(3,4,8,6)(10,13)(14,16)(17,19,22,23)(18,20,24,21)(25,27,29,26)(28,31,32,30);(1,3,5,8)(2,6,7,4)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,24)(22,23);(1,3)(2,4)(5,8)(6,7)(17,20,22,21)(18,19,24,23)(25,28)(26,30)(27,31)(29,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,20)(18,23,24,19)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5394145);