Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,7)(9,10,14,12)(11,16,13,15)(17,18,22,24)(19,20,21,23)(25,26)(27,28)(29,32)(30,31);(1,2,5,6)(3,4,7,8)(17,19,22,21)(18,20,24,23)(25,27)(26,30)(28,32)(29,31);(2,6)(4,8)(9,11,14,13)(10,15,12,16)(17,19,22,21)(18,23,24,20)(25,27)(26,28)(29,31)(30,32);(1,3,5,7)(2,4,6,8)(9,12,14,10)(11,15,13,16)(17,20,22,23)(18,19,24,21);(1,4)(2,7)(3,6)(5,8)(9,13,14,11)(10,15,12,16)(17,21,22,19)(18,23,24,20)(25,28)(26,31)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=g,d^2=gh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4930497);