Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(3,7)(6,8)(10,14)(11,12)(17,18,21,19)(20,23,24,22)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,21,18)(20,23,24,22)(28,36)(31,38)(34,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,11,13,12)(10,15,14,16)(17,20)(18,22)(19,23)(21,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(25,28)(26,31)(27,34)(29,36)(30,37)(32,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5395741);