Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1486 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,12,16)(17,18,22,21)(19,23,20,24)(26,30)(27,33)(32,36)(35,39);(10,14)(15,16)(17,18)(19,24)(20,23)(21,22)(25,26,29,30)(27,31,33,34)(28,32,37,36)(35,38,39,40);(9,11,13,12)(10,15,14,16)(17,19,22,20)(18,23,21,24)(25,26)(27,34)(28,36)(29,30)(31,33)(32,37)(35,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,12,13,11)(10,15,14,16)(17,20,22,19)(18,23,21,24)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(1,3,4,7)(2,5,6,8)(9,10)(11,16)(12,15)(13,14)(17,21,22,18)(19,23,20,24)(25,28,29,37)(26,32,30,36)(27,35,33,39)(31,38,34,40)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=fg,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5497055);