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 | |
1648 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)(11,13)(12,14)(15,16)(17,18)(19,24)(20,22)(21,23)(25,26,30,28)(27,33,35,31)(29,32,38,36)(34,40,37,39);(9,11)(10,13)(12,16)(14,15)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(2,6)(3,8)(10,12)(11,15)(17,19,22,21)(18,23,20,24)(25,27)(26,31)(28,33)(29,34)(30,35)(32,39)(36,40)(37,38);(1,3,5,8)(2,7,6,4)(9,12,14,10)(11,16,15,13)(17,20,22,18)(19,23,21,24)(25,28,30,26)(27,33,35,31)(29,36,38,32)(34,40,37,39);(1,4,5,7)(2,8,6,3)(9,13,14,16)(10,15,12,11)(17,21,22,19)(18,23,20,24)(25,29)(26,32)(27,34)(28,36)(30,38)(31,39)(33,40)(35,37)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fhi,d^2=g,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, 4871281);