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:
(1,2)(3,6)(4,8)(5,7)(9,10,13,14)(11,12,15,16)(25,26)(27,35)(28,38)(29,37)(30,31)(32,36)(33,39)(34,40);(1,3,5,8)(2,6,7,4)(9,11,13,15)(10,12,14,16)(18,21)(22,24)(26,31)(27,36)(33,38)(34,37);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,18,20,21)(19,22,23,24)(25,27)(26,32)(28,37)(29,33)(30,36)(31,35)(34,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19,20,23)(18,22,21,24)(25,28,30,39)(26,33,31,38)(27,37,36,34)(29,35,40,32);(1,4)(2,3)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(25,29)(26,34)(27,38)(28,32)(30,40)(31,37)(33,36)(35,39)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=fghi,d^2=ghi,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, 3460303);