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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2208 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)(3,5)(4,6)(7,8)(10,11)(12,16)(18,21)(19,23)(25,26,29,30)(27,31,32,28);(1,2,4,6)(3,5,7,8)(10,11)(14,15)(17,18)(19,22)(20,21)(23,24)(27,32)(28,31);(3,7)(5,8)(9,10,13,11)(12,14,16,15)(17,18)(19,24)(20,21)(22,23)(25,27)(26,28)(29,32)(30,31);(1,2)(3,8)(4,6)(5,7)(9,11,13,10)(12,15,16,14)(17,18)(19,24)(20,21)(22,23)(25,28)(26,27)(29,31)(30,32);(1,3,4,7)(2,5,6,8)(9,12)(10,14)(11,15)(13,16)(17,19,20,23)(18,22,21,24)(25,26)(27,28)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=fgh,c^2=ghi,d^2=ghi,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, 3461339);