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 | |
1505 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)(3,6)(4,8)(5,7)(9,10,13,15)(11,12,16,14)(19,23)(21,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,10,13,15)(11,12,16,14)(17,18,20,22)(19,21,23,24)(26,27)(29,37)(31,33)(35,38);(2,7)(3,8)(9,11,13,16)(10,14,15,12)(17,18)(19,21)(20,22)(23,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,40,38,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,16)(14,15)(25,28,30,36)(26,31,27,33)(29,38,37,35)(32,40,34,39);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,14)(15,16)(17,19,20,23)(18,21,22,24)(25,29,30,37)(26,32,27,34)(28,35,36,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=fhi,c^2=gi,d^2=f,e^2=fgh,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, 5706846);