Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(9,10,13,15)(11,16,12,14)(17,18,21,23)(19,20,24,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,36)(37,39)(38,40);(1,2,4,5)(3,6,7,8)(9,11)(10,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24)(26,31)(27,34)(33,39)(35,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,20,21,22)(18,19,23,24)(25,27,30,34)(26,32,31,29)(28,35,37,40)(33,36,39,38);(1,2)(3,8)(4,5)(6,7)(9,12,13,11)(10,14,15,16)(17,18)(19,20)(21,23)(22,24)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,23)(21,24)(25,29,30,32)(26,34,31,27)(28,36,37,38)(33,35,39,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=fhi,c^2=gh,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2566692);