Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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 | |
1706 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)(9,10,12,14)(11,15,16,13)(19,20)(22,23)(25,26)(27,32)(28,37)(29,36)(30,31)(33,39)(34,40)(35,38);(1,2,5,4)(3,6,7,8)(9,10,12,14)(11,13,16,15)(17,18,21,24)(19,22,20,23)(26,31)(27,35)(33,37)(34,36);(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27,30,35)(26,32,31,38)(28,36,39,34)(29,33,40,37);(1,3)(2,6)(4,8)(5,7)(17,19,21,20)(18,22,24,23)(25,28,30,39)(26,33,31,37)(27,36,35,34)(29,32,40,38);(1,4,5,2)(3,8,7,6)(9,11,12,16)(10,13,14,15)(17,20,21,19)(18,23,24,22)(25,29,30,40)(26,34,31,36)(27,33,35,37)(28,38,39,32)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=f,d^2=gh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4357889);