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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1767 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,16,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,35)(29,30)(32,36)(33,39)(34,37)(38,40);(1,2,5,3)(4,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18,20,22)(19,23,24,21)(26,29)(31,37)(32,35)(38,39);(2,3)(4,8)(9,11,12,16)(10,13,14,15)(17,19,20,24)(18,21,22,23)(25,27,30,34)(26,31,29,37)(28,33,36,40)(32,38,35,39);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(25,28)(26,32)(27,33)(29,35)(30,36)(31,38)(34,40)(37,39);(1,4,5,8)(2,6,3,7)(9,11)(10,13)(12,16)(14,15)(17,18)(19,23)(20,22)(21,24)(25,29,30,26)(27,31,34,37)(28,35,36,32)(33,38,40,39)Polycyclic presentation:
2^9|a^2=fghi,b^2=fgi,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, 5397722);