Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,6)(4,8)(11,15)(14,16)(25,26,29,30)(27,31,33,34)(28,32,36,38)(35,39,37,40);(1,2,5,6)(3,4,7,8)(17,18,21,20)(19,23,24,22)(26,30)(27,33)(28,36)(39,40);(2,6)(3,7)(9,10,13,12)(11,14,15,16)(17,19,21,24)(18,22,20,23)(25,26)(27,34)(28,32)(29,30)(31,33)(35,40)(36,38)(37,39);(1,3,5,7)(2,4,6,8)(9,11,13,15)(10,14,12,16)(17,18,21,20)(19,22,24,23)(25,27)(26,31)(28,37)(29,33)(30,34)(32,40)(35,36)(38,39);(1,4)(2,3)(5,8)(6,7)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,23,24,22)(25,28,29,36)(26,32,30,38)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fhi,d^2=ghi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3238749);