Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,12)(13,14)(15,16)(17,18,20,22)(19,23,24,21)(25,26)(27,30)(28,36)(29,32)(31,37)(33,35)(34,39)(38,40);(2,5)(3,8)(9,10)(11,16)(12,15)(13,14)(19,24)(21,23)(25,26,29,32)(27,30,35,33)(28,36,37,31)(34,39,40,38);(3,8)(6,7)(10,14)(11,15)(17,19,20,24)(18,21,22,23)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,37)(38,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,15)(10,12,14,16)(17,18)(19,23)(20,22)(21,24)(25,27)(26,30)(28,34)(29,35)(31,38)(32,33)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,15)(11,14)(13,16)(25,28,29,37)(26,31,32,36)(27,34,35,40)(30,38,33,39)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=fghi,d^2=fhi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6038171);