Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1368 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(9,10,14,12)(11,15,13,16)(18,21)(22,24)(25,26)(27,28)(29,37)(30,31)(32,36)(33,39)(34,38)(35,40);(1,2,5,4)(3,6,7,8)(9,11,14,13)(10,15,12,16)(19,23)(22,24)(26,31)(27,32)(33,37)(34,40);(9,12,14,10)(11,15,13,16)(17,18,20,21)(19,22,23,24)(25,27,30,32)(26,28,31,36)(29,34,39,40)(33,35,37,38);(1,3)(2,6)(4,8)(5,7)(9,11,14,13)(10,16,12,15)(17,19)(18,22)(20,23)(21,24)(25,28)(26,32)(27,31)(29,38)(30,36)(33,34)(35,39)(37,40);(1,4,5,2)(3,8,7,6)(9,13,14,11)(10,15,12,16)(25,29,30,39)(26,33,31,37)(27,34,32,40)(28,35,36,38)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=gh,d^2=fh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4827039);