Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,6)(3,7,8,4)(11,15)(14,16)(20,24)(22,23)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,40)(38,39);(1,2)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18,21,19)(20,22,24,23)(26,31)(27,32)(33,39)(34,37);(2,6)(4,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,23,24,22)(25,27,30,32)(26,28,31,36)(29,37,38,34)(33,40,39,35);(1,3)(2,7)(4,6)(5,8)(17,20,21,24)(18,22,19,23)(25,28)(26,32)(27,31)(29,35)(30,36)(33,37)(34,39)(38,40);(1,4,5,7)(2,3,6,8)(9,11)(10,14)(12,15)(13,16)(25,29,30,38)(26,33,31,39)(27,34,32,37)(28,35,36,40)Polycyclic presentation:
2^9|a^2=g,b^2=fgi,c^2=i,d^2=fgi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4621398);