Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1676 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,7)(3,4,8,6)(11,15)(13,16)(19,23)(22,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,40)(36,37)(38,39);(1,3,5,8)(2,6,7,4)(9,10,12,14)(11,13,15,16)(18,21)(22,24)(26,31)(27,33)(34,39)(35,37);(2,7)(4,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,21)(23,24)(25,27,30,33)(26,32,31,28)(29,37,38,35)(34,36,39,40);(1,3)(2,4)(5,8)(6,7)(17,19)(18,22)(20,23)(21,24)(25,28,30,32)(26,33,31,27)(29,36,38,40)(34,37,39,35);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,38)(26,34,31,39)(27,35,33,37)(28,36,32,40)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=fg,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4523578);