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 | |
1540 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(11,15)(14,16)(25,26,31,34)(27,29,28,32)(30,33,39,40)(35,37,36,38);(2,3)(6,7)(18,21)(22,24)(25,27,31,28)(26,32,34,29)(30,35,39,36)(33,38,40,37);(1,2,5,3)(4,6,8,7)(9,10,13,12)(11,14,15,16)(17,18,20,21)(19,22,23,24)(25,28,31,27)(26,32,34,29)(30,35,39,36)(33,37,40,38);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,14,12,16)(25,29,31,32)(26,27,34,28)(30,38,39,37)(33,36,40,35);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,16,15,14)(17,19,20,23)(18,22,21,24)(25,30,31,39)(26,33,34,40)(27,35,28,36)(29,37,32,38)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=ghi,d^2=i,e^2=hi,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, 3720300);