Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,13,12)(11,15,16,14)(19,23)(22,24)(25,26)(27,32)(28,34)(29,37)(30,31)(33,38)(35,36)(39,40);(1,2)(3,5)(4,7)(6,8)(9,11,13,16)(10,14,12,15)(18,21)(22,24)(26,31)(27,34)(29,38)(36,39);(2,3)(6,7)(10,12)(14,15)(17,18)(19,22)(20,21)(23,24)(25,27,30,34)(26,32,31,28)(29,35,38,40)(33,39,37,36);(1,3,5,2)(4,7,8,6)(9,12,13,10)(11,14,16,15)(17,19)(18,22)(20,23)(21,24)(25,28,30,32)(26,27,31,34)(29,36,38,39)(33,35,37,40);(1,4)(2,6)(3,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,29)(26,33)(27,35)(28,36)(30,38)(31,37)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=fg,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4382082);