Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,16)(12,14)(13,15)(18,21)(22,24)(27,31)(29,32);(3,8)(5,7)(9,11,13,14)(10,12,15,16)(19,23)(22,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,16)(12,14)(13,15)(17,18)(19,22)(20,21)(23,24)(25,26,28,30)(27,29,31,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,29)(28,31)(30,32);(1,2)(3,5)(4,6)(7,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2537192);