Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| 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 | |
2020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,13)(14,16)(18,20)(21,23)(25,26,29,31)(27,32,28,30);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(17,18,22,20)(19,21,24,23)(25,27,29,28)(26,30,31,32);(9,11)(10,14)(12,16)(13,15)(17,19,22,24)(18,23,20,21)(25,28,29,27)(26,30,31,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,18)(19,21,24,23);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,21)(18,19)(20,24)(22,23)(25,26)(27,32)(28,30)(29,31)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fghi,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1960460);