Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,7,8)(9,10)(11,12)(13,14)(15,16)(18,22)(19,24)(26,30)(31,32);(2,4)(6,8)(9,10,13,14)(11,12,15,16);(3,7)(6,8)(10,14)(11,15)(17,18,21,22)(19,23,24,20)(25,26)(27,31)(28,32)(29,30);(1,3,5,7)(2,6,4,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,23)(20,22)(21,24)(25,27,29,28)(26,31,30,32);(1,4,5,2)(3,8,7,6)(9,12)(10,11)(13,16)(14,15)(17,20,21,23)(18,19,22,24)(25,28,29,27)(26,32,30,31)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=i,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 639661);