Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437798 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3008 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,8)(4,6)(5,7)(9,10,13,18,14,19,27,15)(11,16,23,32,22,30,20,28)(12,17,25,34,26,33,24,29)(21,31,37,40,36,39,35,38);(10,15)(11,20)(12,24)(13,27)(16,30)(17,33)(18,19)(22,23)(25,26)(31,38)(35,37)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,21)(13,23)(14,22)(15,28)(17,31)(18,32)(19,30)(20,27)(24,35)(25,37)(26,36)(29,38)(33,39)(34,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21)(13,25)(14,26)(15,29)(16,31)(18,34)(19,33)(20,35)(22,36)(23,37)(24,27)(28,38)(30,39)(32,40);(11,22)(16,30)(20,23)(21,36)(28,32)(31,39)(35,37)(38,40)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359227);