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 | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,4,7)(10,14)(11,16)(18,23)(19,21)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,2)(3,8)(4,7)(5,6)(9,10,13,14)(11,15,16,12)(17,18,22,23)(19,24,21,20)(26,30)(31,33)(32,38)(39,40);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,16)(14,15)(17,19,22,21)(18,24,23,20)(25,26)(27,33)(28,32)(29,30)(31,35)(34,40)(36,39)(37,38);(1,4,6,3)(2,7,5,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,19)(21,23)(22,24)(25,27,29,35)(26,31,30,33)(28,36,37,34)(32,40,38,39);(1,5,6,2)(3,8,4,7)(9,12,13,15)(10,16,14,11)(17,21,22,19)(18,24,23,20)(25,28,29,37)(26,32,30,38)(27,34,35,36)(31,39,33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,c^2=fi,d^2=fghi,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, 5841343);