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) = C25×C4 |
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 | |
1530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(6,8)(9,10)(11,12)(13,15)(14,16)(19,23)(22,24)(25,26,30,32)(27,33,35,28)(29,31,38,40)(34,39,37,36);(9,11)(10,14)(12,15)(13,16)(18,21)(22,24)(25,27,30,35)(26,28,32,33)(29,37,38,34)(31,39,40,36);(11,16)(12,14)(17,18,20,21)(19,22,23,24)(25,27)(26,28)(29,37)(30,35)(31,39)(32,33)(34,38)(36,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,14)(15,16)(25,28)(26,27)(29,36)(30,33)(31,34)(32,35)(37,40)(38,39);(1,3,4,7)(2,6,5,8)(9,11)(10,12)(13,16)(14,15)(17,19,20,23)(18,22,21,24)(25,29)(26,31)(27,34)(28,36)(30,38)(32,40)(33,39)(35,37)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fh,e^2=fhi,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, 5270479);