Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,21,20)(19,22,23,24)(25,26,30,31)(27,32,34,28)(29,33,39,40)(35,38,37,36);(9,10,13,11)(12,15,16,14)(19,23)(22,24)(26,31)(27,34)(33,40)(35,37);(10,11)(12,16)(17,19)(18,22)(20,24)(21,23)(25,27)(26,32)(28,31)(29,37)(30,34)(33,36)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,14,16,15)(25,28,30,32)(26,27,31,34)(29,38,39,36)(33,37,40,35);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,20,21,18)(19,24,23,22)(25,29,30,39)(26,33,31,40)(27,35,34,37)(28,36,32,38)Polycyclic presentation:
2^9|a^2=gh,b^2=g,d^2=f,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4171325);