Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | |
1794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,20,23,21)(26,31)(27,29);(17,19,22,23)(18,21,24,20)(25,26,30,31)(27,28,29,32);(10,11)(14,15)(17,20,22,21)(18,23,24,19)(25,27,30,29)(26,32,31,28);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,28,30,32)(26,27,31,29);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,21,22,20)(18,19,24,23)(25,29,30,27)(26,32,31,28)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=f,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4838646);