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 | |
1494 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)(3,4)(5,6)(7,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,21)(22,23)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,38,40,39);(2,6)(3,8)(9,11,13,16)(10,12,14,15)(17,19,22,21)(18,20,23,24)(26,27)(31,33)(32,34)(38,39);(2,6)(4,7)(10,14)(12,15)(18,23)(19,21)(25,27,30,26)(28,33,36,31)(29,32,37,34)(35,38,40,39);(1,3,5,8)(2,7,6,4)(9,11)(10,15)(12,14)(13,16)(17,20)(18,19)(21,23)(22,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39);(1,4,5,7)(2,8,6,3)(9,12,13,15)(10,11,14,16)(17,21,22,19)(18,20,23,24)(25,29,30,37)(26,32,27,34)(28,35,36,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=fhi,c^2=fg,d^2=hi,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, 4593467);