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 | |
1482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(4,8)(9,10,13,12)(11,14,15,16)(17,18,21,19)(20,23,24,22)(25,26,30,29)(27,33,34,31)(28,36,37,32)(35,39,38,40);(2,3)(6,7)(11,15)(14,16)(17,19,21,18)(20,23,24,22)(26,29)(27,34)(28,37)(39,40);(1,2,5,3)(4,6,8,7)(9,11,13,15)(10,14,12,16)(25,27,30,34)(26,31,29,33)(28,38,37,35)(32,40,36,39);(1,3,5,2)(4,7,8,6)(9,10,13,12)(11,14,15,16)(17,20,21,24)(18,22,19,23)(25,28,30,37)(26,32,29,36)(27,35,34,38)(31,39,33,40);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,16,15,14)(25,29,30,26)(27,33,34,31)(28,32,37,36)(35,39,38,40)Polycyclic presentation:
2^9|a^2=fhi,b^2=gi,c^2=h,d^2=ghi,e^2=fgh,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, 3075482);