Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1416 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,16,12)(25,26)(27,28)(29,37)(30,31)(32,34)(33,40)(35,39)(36,38);(1,2,4,6)(3,5,7,8)(10,14)(11,16)(18,22)(23,24)(26,31)(27,34)(33,37)(35,38);(3,7)(5,8)(9,11,13,16)(10,15,14,12)(17,18,21,22)(19,23,20,24)(25,27)(26,32)(28,31)(29,38)(30,34)(33,36)(35,40)(37,39);(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,20)(18,23,22,24)(25,28,30,32)(26,27,31,34)(29,39,40,36)(33,38,37,35);(1,3)(2,5)(4,7)(6,8)(9,12,13,15)(10,11,14,16)(17,20,21,19)(18,24,22,23)(25,29,30,40)(26,33,31,37)(27,35,34,38)(28,36,32,39)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=gh,d^2=hi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3435179);