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 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,6,4,7)(9,10,12,13)(11,15,16,14)(17,18,20,22)(19,23,24,21)(25,26)(27,31)(28,30)(29,32)(33,34)(35,39)(36,38)(37,40);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(26,28)(27,33)(32,36)(35,37);(10,13)(14,15)(17,18)(19,23)(20,22)(21,24)(25,27,30,33)(26,31,28,34)(29,37,38,35)(32,40,36,39);(1,2)(3,6)(4,7)(5,8)(9,11,12,16)(10,14,13,15)(25,28,30,26)(27,34,33,31)(29,36,38,32)(35,40,37,39);(1,4,5,3)(2,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,21)(20,24)(22,23)(25,29,30,38)(26,32,28,36)(27,35,33,37)(31,39,34,40)Polycyclic presentation:
2^9|a^2=ghi,b^2=hi,c^2=fg,d^2=h,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, 2431165);