Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10,12,13)(11,15,16,14)(17,18,21,20)(19,22,24,23)(25,26)(27,29)(28,35)(30,32)(31,33)(34,37)(36,39)(38,40);(10,13)(11,16)(17,18)(19,23)(20,21)(22,24)(25,27,31,28)(26,29,33,35)(30,36,38,34)(32,39,40,37);(10,13)(14,15)(18,20)(22,23)(25,28,31,27)(26,29,33,35)(30,36,38,34)(32,37,40,39);(1,2)(3,5)(4,6)(7,8)(9,11,12,16)(10,14,13,15)(17,19)(18,22)(20,23)(21,24)(25,29)(26,28)(27,33)(30,37)(31,35)(32,36)(34,40)(38,39);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,23,24,22)(25,30,31,38)(26,32,33,40)(27,34,28,36)(29,37,35,39)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=fg,d^2=fh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5292122);