Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,8)(3,5,4,7)(11,15)(14,16)(17,18,21,22)(19,20,24,23)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(1,3,6,4)(2,7,8,5)(18,22)(20,23)(27,34)(30,33)(35,40)(38,39);(1,3,6,4)(2,5,8,7)(9,10,13,12)(11,14,15,16)(17,18,21,22)(19,23,24,20)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,37)(39,40);(1,4,6,3)(2,7,8,5)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24)(25,27,29,34)(26,30,32,33)(28,35,37,40)(31,38,36,39);(1,5,6,7)(2,3,8,4)(9,12,13,10)(11,16,15,14)(17,20,21,23)(18,19,22,24)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=fghi,d^2=fgh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1983554);