Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #845297 |
| 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 | |
2322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,12)(13,15)(14,16)(17,18,22,24)(19,23,21,20)(25,26,29,30)(27,28,31,32);(2,5)(6,7)(9,11,13,16)(10,12,15,14)(17,19,22,21)(18,20,24,23)(26,30)(27,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,23)(18,19,24,21)(25,27)(26,28)(29,31)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,20)(21,23)(22,24);(1,3)(2,6)(4,8)(5,7)(9,12,13,14)(10,11,15,16)(17,21,22,19)(18,23,24,20)(25,28)(26,27)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 768720);