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 | |
1820 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(17,18,21,24)(19,20,23,22)(25,26,29,30)(27,31,35,33)(28,32,37,36)(34,38,40,39);(1,2,5,7)(3,6,8,4)(11,15)(13,16)(17,19,21,23)(18,22,24,20)(26,30)(31,33)(32,36)(38,39);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,4)(2,3)(5,6)(7,8)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,23)(19,24)(21,22)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=i,e^2=g,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, 5398565);