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 | |
1632 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)(11,14)(12,13)(15,16)(25,26,30,33)(27,34,31,28)(29,37,38,32)(35,36,39,40);(1,2,4,5)(3,6,7,8)(10,13)(14,15)(18,21)(22,24)(27,31)(28,34)(35,39)(36,40);(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,21)(23,24)(25,27,30,31)(26,28,33,34)(29,35,38,39)(32,36,37,40);(1,2)(3,8)(4,5)(6,7)(9,10)(11,15)(12,13)(14,16)(17,19,20,23)(18,22,21,24)(25,28)(26,31)(27,33)(29,36)(30,34)(32,39)(35,37)(38,40);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(25,29)(26,32)(27,35)(28,36)(30,38)(31,39)(33,37)(34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=g,d^2=fhi,e^2=f,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, 3408481);