Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1418353 |
| automorphism | ? |
| 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 | |
2085 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,7)(3,6,8,5)(10,12)(11,16)(17,18)(19,21)(20,22)(23,24)(25,26,29,32)(27,31,28,30);(1,3,4,8)(2,5,7,6)(9,10)(11,15)(12,13)(14,16)(17,19)(18,20)(21,24)(22,23);(9,11)(10,14)(12,15)(13,16)(17,20,23,21)(18,19,24,22)(25,27,29,28)(26,30,32,31);(1,2)(3,5)(4,7)(6,8)(9,11)(10,15)(12,14)(13,16)(17,21,23,20)(18,19,24,22)(25,26)(27,30)(28,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,15,16,14)(17,22,23,19)(18,21,24,20)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fi,d^2=fghi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1406733);