Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1578 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(9,10,13,15)(11,16,12,14)(18,21)(19,23)(27,31)(30,32);(9,11,13,12)(10,14,15,16)(17,18)(19,24)(20,21)(22,23)(26,29)(30,32);(11,12)(14,16)(17,18,20,21)(19,22,23,24)(25,26)(27,30)(28,29)(31,32);(1,3,5,8)(2,6,4,7)(9,12,13,11)(10,14,15,16)(17,18)(19,22)(20,21)(23,24);(1,4,5,2)(3,7,8,6)(9,10)(11,16)(12,14)(13,15)(17,19,20,23)(18,22,21,24)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=g,d^2=fi,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4285709);