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 | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,16,12)(17,18)(19,24)(20,21)(22,23)(26,28)(27,31);(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,16)(13,14)(18,23)(19,20)(25,26,29,28)(27,30,31,32);(3,7)(5,8)(10,14)(12,15)(17,19,22,20)(18,24,23,21)(25,27)(26,30)(28,32)(29,31);(1,2)(3,8)(4,6)(5,7)(9,11)(10,15)(12,14)(13,16)(17,20,22,19)(18,21,23,24)(25,27)(26,30)(28,32)(29,31);(1,3,4,7)(2,5,6,8)(9,12,13,15)(10,16,14,11)(17,21)(18,20)(19,23)(22,24)(25,28,29,26)(27,30,31,32)Polycyclic presentation:
2^9|a^2=gi,b^2=fh,c^2=ghi,d^2=ghi,e^2=fghi,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, 3764123);