Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(11,15)(13,16)(19,23)(20,22)(25,26)(27,31)(28,32)(29,30)(33,38)(34,39)(35,37)(36,40);(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,20,23,22)(26,30)(27,33)(32,35)(34,36);(1,2,5,4)(3,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,23)(18,22,24,20)(25,26)(27,31)(28,35)(29,30)(32,37)(33,38)(34,40)(36,39);(1,3,5,7)(2,6,4,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27,29,33)(26,31,30,38)(28,36,37,34)(32,40,35,39);(1,4,5,2)(3,8,7,6)(9,11)(10,13)(12,15)(14,16)(17,20,21,22)(18,23,24,19)(25,28)(26,32)(27,34)(29,37)(30,35)(31,39)(33,36)(38,40)Polycyclic presentation:
2^9|b^2=g,c^2=fhi,d^2=fi,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3720261);