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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,16,14,12)(19,23)(21,24)(25,26,30,33)(27,29,34,31)(28,36,39,32)(35,40,37,38);(3,7)(6,8)(9,11,13,14)(10,12,15,16)(27,34)(29,31)(35,37)(38,40);(2,5)(6,8)(9,12,13,16)(10,14,15,11)(17,18,20,22)(19,21,23,24)(25,27)(26,31)(28,37)(29,33)(30,34)(32,38)(35,39)(36,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,14)(15,16)(17,19,20,23)(18,21,22,24)(25,28,30,39)(26,32,33,36)(27,35,34,37)(29,38,31,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,16)(14,15)(25,29,30,31)(26,27,33,34)(28,38,39,40)(32,35,36,37)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fhi,d^2=fi,e^2=fh,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, 5238649);