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 | |
1574 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)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,26)(27,32)(28,34)(29,33)(30,31)(35,37)(36,40)(38,39);(1,2,5,3)(4,7,8,6)(11,15)(13,16)(17,19)(18,22)(20,23)(21,24)(26,31)(27,34)(33,39)(35,40);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(17,19,21,24)(18,23,20,22)(25,27,30,34)(26,32,31,28)(29,35,38,40)(33,37,39,36);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,22)(21,24)(25,28,30,32)(26,27,31,34)(29,37,38,36)(33,40,39,35);(1,4,5,8)(2,6,3,7)(9,11)(10,13)(12,15)(14,16)(17,20,21,18)(19,22,24,23)(25,29)(26,33)(27,35)(28,36)(30,38)(31,39)(32,37)(34,40)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=fghi,d^2=fhi,e^2=fh,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, 5861631);