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 | |
1553 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)(11,16)(12,15)(13,14)(20,24)(22,23)(25,26,29,32)(27,33,34,30)(28,31,36,38)(35,39,40,37);(2,3)(6,7)(9,11,14,12)(10,15,13,16)(17,18,21,19)(20,22,24,23)(27,34)(30,33)(35,40)(37,39);(1,2,5,3)(4,7,8,6)(9,12,14,11)(10,15,13,16)(17,19,21,18)(20,23,24,22)(25,26)(27,30)(28,31)(29,32)(33,34)(35,37)(36,38)(39,40);(1,3,5,2)(4,7,8,6)(17,20,21,24)(18,22,19,23)(25,27)(26,30)(28,35)(29,34)(31,37)(32,33)(36,40)(38,39);(1,4,5,8)(2,6,3,7)(9,13,14,10)(11,16,12,15)(25,28,29,36)(26,31,32,38)(27,35,34,40)(30,37,33,39)Polycyclic presentation:
2^9|a^2=gh,b^2=fghi,c^2=fgi,d^2=ghi,e^2=fgh,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, 4637394);