Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5020720 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(14,18)(19,21)(20,24)(26,27)(29,30,35,32)(31,38,36,33)(34,40,42,37)(39,41,43,44);(13,14,17,18)(15,19,23,21)(16,20,25,24)(22,26,28,27)(29,31)(30,33)(32,38)(34,39)(35,36)(37,41)(40,44)(42,43);(6,9)(10,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,26)(27,28)(29,32,35,30)(31,38,36,33)(34,37,42,40)(39,41,43,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,23)(14,19,18,21)(16,22,25,28)(20,26,24,27)(29,33,35,38)(30,36,32,31)(34,41,42,44)(37,43,40,39);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28)(29,34,35,42)(30,37,32,40)(31,39,36,43)(33,41,38,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=fh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4928652);