Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,8,4)(9,10)(11,13)(12,14)(15,16)(17,18,23,24)(19,20,22,21)(27,31)(29,32);(1,3,5,8)(2,4,7,6)(10,11)(12,16)(17,19)(18,21)(20,24)(22,23);(3,8)(4,6)(9,10,13,11)(12,15,16,14)(17,20,23,21)(18,19,24,22)(25,26,28,30)(27,29,31,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,10)(12,15,16,14)(17,21,23,20)(18,19,24,22)(25,27,28,31)(26,29,30,32);(1,4,5,6)(2,3,7,8)(9,12,13,16)(10,14,11,15)(17,22,23,19)(18,20,24,21)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fhi,d^2=fhi,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, 5230553);