Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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 | |
1744 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)(3,5)(4,6)(7,8)(9,10,14,16)(11,12,13,15)(17,18,21,23)(19,24,22,20)(25,26,29,28)(27,30,31,32);(1,2,4,6)(3,5,7,8)(9,11)(10,15)(12,16)(13,14)(19,22)(20,24)(27,31)(30,32);(3,7)(5,8)(9,12,14,15)(10,13,16,11)(17,19,21,22)(18,20,23,24)(25,27,29,31)(26,30,28,32);(1,2)(3,8)(4,6)(5,7)(9,13,14,11)(10,12,16,15)(25,28,29,26)(27,32,31,30);(1,3)(2,5)(4,7)(6,8)(9,12)(10,13)(11,16)(14,15)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=h,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3964475);