Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #16502 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2481 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(3,8)(9,10)(11,15)(12,16)(13,14)(17,18,20,22)(19,23,24,21)(25,26)(27,28)(29,30)(31,32);(1,2,5,6)(3,4,8,7)(9,11)(10,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,27)(26,31)(28,29)(30,32);(2,6)(4,7)(10,13)(11,16)(17,18)(19,23)(20,22)(21,24)(25,28,30,31)(26,32,29,27);(1,3)(2,7)(4,6)(5,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,23)(20,22)(21,24)(25,27)(26,28)(29,31)(30,32);(1,4,5,7)(2,8,6,3)(9,13,14,10)(11,15,16,12)(17,19,20,24)(18,21,22,23)(25,29,30,26)(27,28,32,31)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3306560);