Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1452 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(9,11,14,13)(10,15,12,16)(18,19)(20,24)(27,33)(31,34)(35,39)(38,40);(2,6)(3,8)(10,12)(15,16)(17,19,21,18)(20,23,24,22)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40);(1,3,5,8)(2,7,6,4)(9,12,14,10)(11,16,13,15)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,4,5,7)(2,8,6,3)(9,13,14,11)(10,16,12,15)(17,20)(18,22)(19,23)(21,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=h,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4861731);