Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,8)(11,15)(14,16)(17,18,21,19)(20,23,24,22)(25,26,30,31)(27,32,34,29)(28,33,37,39)(35,36,40,38);(1,2)(3,6)(4,5)(7,8)(10,13)(14,16)(17,19,21,18)(20,23,24,22)(26,31)(27,34)(33,39)(35,40);(1,3)(2,6)(4,8)(5,7)(9,10,12,13)(11,14,15,16)(25,27)(26,32)(28,35)(29,31)(30,34)(33,36)(37,40)(38,39);(1,3)(2,7)(4,8)(5,6)(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,2)(3,6)(4,5)(7,8)(25,29,30,32)(26,27,31,34)(28,36,37,38)(33,40,39,35)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=fgh,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2566695);