Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(7,11)(9,12)(13,14)(15,21)(16,18)(17,20)(19,23)(22,28)(24,26)(25,27)(32,39)(35,42)(37,43)(40,44);(14,16)(19,21)(20,24)(27,28)(29,30,33,36)(31,34,38,41)(32,35,39,42)(37,40,43,44);(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(5,7,8,11)(6,9,10,12)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,31,33,38)(30,34,36,41)(32,37,39,43)(35,40,42,44);(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,32,33,39)(30,35,36,42)(31,37,38,43)(34,40,41,44)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=i,d^2=hi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4304861);