Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1708 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)(3,5,7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,23)(20,24)(21,22)(25,26,29,30)(27,31,32,28);(9,11,13,16)(10,14,15,12)(17,19,22,20)(18,23,21,24)(26,30)(27,32);(9,12)(10,11)(13,14)(15,16)(17,18,22,21)(19,24,20,23)(25,27,29,32)(26,31,30,28);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,20,22,19)(18,23,21,24)(25,26)(27,31)(28,32)(29,30);(1,3,4,7)(2,5,6,8)(9,11)(10,12)(13,16)(14,15)(17,21,22,18)(19,23,20,24)(25,28,29,31)(26,27,30,32)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=h,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4028731);