Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 4194304 |
| 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 | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2108 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)(3,7)(4,5)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18,21,20)(19,23,24,22)(25,26)(27,28)(29,31)(30,32);(2,4)(3,8)(10,14)(11,16)(17,18)(19,23)(20,21)(22,24)(25,27,30,29)(26,31,32,28);(2,4)(6,7)(11,16)(12,15)(18,20)(22,23)(27,29)(28,31);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,15,14,12)(17,19)(18,22)(20,23)(21,24)(25,28,30,31)(26,27,32,29);(1,4,5,2)(3,7,8,6)(9,12,13,15)(10,16,14,11)(17,20,21,18)(19,23,24,22)(25,29,30,27)(26,31,32,28)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5143673);