Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| 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 | |
1594 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,21)(22,24)(25,26,29,30)(27,33,34,31)(28,32,36,37)(35,39,40,38);(2,4)(6,7)(10,13)(15,16)(18,21)(19,24)(26,30)(27,34)(28,36)(38,39);(9,11,14,12)(10,15,13,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,38)(36,37)(39,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,11)(10,16,13,15)(25,27,29,34)(26,31,30,33)(28,35,36,40)(32,38,37,39);(1,4,5,2)(3,7,8,6)(9,13,14,10)(11,16,12,15)(17,19)(18,22)(20,24)(21,23)(25,28,29,36)(26,32,30,37)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,c^2=fghi,d^2=fhi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1980791);