Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1586 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)(3,6)(4,5)(7,8)(10,13)(14,15)(17,18)(19,23)(20,21)(22,24)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38);(9,10)(11,15)(12,13)(14,16)(17,19,22,21)(18,23,24,20)(27,33)(30,37)(34,39)(38,40);(2,5)(6,8)(10,13)(14,15)(18,24)(20,23)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,24)(22,23)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(17,21,22,19)(18,20,24,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=fh,b^2=hi,c^2=fh,d^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4862543);