Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31);(9,11,12,15)(10,13,14,16)(18,23)(21,25)(24,31)(27,32);(5,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,32)(26,30);(1,2)(3,4)(9,10)(11,13)(12,14)(15,16)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32);(17,21,22,25)(18,20,23,30)(19,27,28,32)(24,29,31,26)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,e^2=f,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, 594512);