Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,8,4,5)(10,13)(14,16)(18,22)(19,20)(25,26)(27,29)(28,30)(31,32);(1,3)(2,5)(4,6)(7,8)(11,15)(14,16)(17,18,21,22)(19,23,20,24)(25,27,28,31)(26,29,30,32);(3,4)(5,8)(9,10,12,13)(11,14,15,16)(17,19,21,20)(18,23,22,24)(25,26,28,30)(27,29,31,32);(1,4,6,3)(2,5,7,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,23,22,24)(25,26)(27,29)(28,30)(31,32);(1,5)(2,3)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,24)(20,23)(21,22)(25,27)(26,29)(28,31)(30,32)Polycyclic presentation:
2^9|a^2=h,b^2=fghi,c^2=g,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5135934);