Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(11,15)(14,16)(19,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,40)(37,38);(1,2,5,3)(4,7,8,6)(10,13)(14,16)(18,21)(22,24)(26,30)(31,33)(32,38)(39,40);(2,3)(6,7)(9,10,12,13)(11,14,15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,3,5,2)(4,7,8,6)(17,19)(18,22)(20,23)(21,24)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(1,4)(2,6)(3,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fghi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5594015);