Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1430 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)(3,4)(5,8)(6,7)(11,15)(14,16)(17,18,21,20)(19,22,23,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(1,3)(2,5)(4,6)(7,8)(19,23)(22,24)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(1,4,7,5)(2,8,6,3)(9,10,13,12)(11,14,15,16)(17,19,21,23)(18,22,20,24)(25,28,31,27)(26,32,29,34)(30,35,38,36)(33,40,37,39);(1,5,7,4)(2,8,6,3)(9,11)(10,14)(12,16)(13,15)(25,29,31,26)(27,32,28,34)(30,37,38,33)(35,39,36,40);(1,6,7,2)(3,4,8,5)(9,12,13,10)(11,16,15,14)(17,20,21,18)(19,24,23,22)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=hi,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2439299);