Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(3,8)(9,10)(11,15)(12,16)(13,14)(20,24)(22,23)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(1,2)(3,7)(4,8)(5,6)(9,11,13,16)(10,12,14,15)(17,18,21,19)(20,22,24,23)(27,33)(30,37)(34,39)(38,40);(2,6)(4,7)(9,10,13,14)(11,15,16,12)(17,19,21,18)(20,23,24,22)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(1,3,5,8)(2,7,6,4)(9,12)(10,11)(13,15)(14,16)(17,20,21,24)(18,22,19,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(1,4,5,7)(2,8,6,3)(9,10)(11,12)(13,14)(15,16)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)Polycyclic presentation:
2^9|a^2=fh,b^2=ghi,c^2=fgi,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4637353);