Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,13,14)(11,15,16,12)(17,18)(19,23)(20,24)(21,22)(26,28)(27,33)(29,37)(39,40);(3,8)(5,7)(10,14)(11,16)(18,22)(19,24)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,38,39);(11,16)(12,15)(17,19,21,24)(18,23,22,20)(25,27)(26,31)(28,34)(29,38)(30,33)(32,40)(35,37)(36,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,22)(21,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,38,39);(1,2)(3,5)(4,6)(7,8)(9,12,13,15)(10,11,14,16)(17,20,21,23)(18,19,22,24)(25,29)(26,32)(27,35)(28,36)(30,37)(31,39)(33,38)(34,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=h,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6075089);