Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #616960 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15763 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,13)(12,15)(14,16)(17,18,21,22)(19,24,20,23)(25,26)(27,31)(28,30)(29,32);(1,3,5,8)(2,6,7,4)(10,11)(14,15)(18,22)(23,24)(26,28)(27,32);(9,10,13,11)(12,14,16,15)(17,19,21,20)(18,23,22,24)(25,27,30,32)(26,31,28,29);(1,3)(2,4)(5,8)(6,7)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,24)(21,22)(25,28,30,26)(27,29,32,31);(1,4)(2,3)(5,6)(7,8)(9,12,13,16)(10,14,11,15)(17,20,21,19)(18,24,22,23)(25,29)(26,27)(28,32)(30,31)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=i,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 532169);