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 | |
1894 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(9,12)(13,14)(15,18)(16,19)(17,22)(20,25)(21,23)(24,27)(26,28)(29,30,34,35)(31,32,36,39)(33,37,42,43)(38,40,41,44);(5,6,8,10)(7,9,11,12)(14,15)(16,23)(20,22)(24,28)(30,35)(32,39)(37,43)(40,44);(1,2)(3,4)(13,15,18,14)(16,21,23,19)(17,20,25,22)(24,26,28,27)(29,31,34,36)(30,32,35,39)(33,41,42,38)(37,44,43,40);(5,7)(6,9)(8,11)(10,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,26)(22,27)(25,28)(29,32)(30,36)(31,35)(33,40)(34,39)(37,41)(38,43)(42,44);(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)(29,33)(30,37)(31,38)(32,40)(34,42)(35,43)(36,41)(39,44)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=f,e^2=h,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, 4292715);