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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1442 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,4)(3,8,5,7)(9,10)(11,14)(12,13)(15,16)(17,18,21,19)(20,23,24,22)(25,26,30,33)(27,34,32,29)(28,36,39,31)(35,38,37,40);(1,2,6,4)(3,7,5,8)(10,13)(11,15)(17,19,21,18)(20,23,24,22)(27,32)(29,34)(35,37)(38,40);(1,3,6,5)(2,7,4,8)(9,11,12,15)(10,14,13,16)(25,27,30,32)(26,29,33,34)(28,37,39,35)(31,40,36,38);(1,4,6,2)(3,7,5,8)(17,20)(18,22)(19,23)(21,24)(25,28,30,39)(26,31,33,36)(27,35,32,37)(29,38,34,40);(1,5,6,3)(2,7,4,8)(9,10)(11,16)(12,13)(14,15)(25,29)(26,32)(27,33)(28,38)(30,34)(31,37)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=gi,c^2=fg,d^2=g,e^2=fgh,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, 5397720);