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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1554 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)(3,6)(4,5)(7,8)(10,13)(11,15)(17,18,22,24)(19,20,21,23)(28,35)(32,37)(33,39)(38,40);(3,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,21)(18,23,24,20)(26,30)(31,36)(32,37)(38,40);(2,5)(6,8)(11,15)(14,16)(17,20,22,23)(18,21,24,19)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,19)(18,23,24,20)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,16)(12,13)(14,15)(17,18)(19,23)(20,21)(22,24)(25,28,29,35)(26,32,30,37)(27,33,34,39)(31,38,36,40)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fhi,e^2=h,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, 5854322);