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×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2650 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)(9,10,15,21,16,22,14,20)(11,23,26,35,27,36,32,17)(12,18,29,34,25,38,30,37)(13,19,28,33,24,40,31,39);(9,11)(10,17)(12,28)(13,29)(14,26)(15,32)(16,27)(18,19)(20,23)(21,36)(22,35)(24,30)(25,31)(33,37)(34,39)(38,40);(1,3,5,4)(2,6,7,8)(9,12)(10,18)(11,24)(13,27)(14,30)(15,29)(16,25)(17,33)(19,36)(20,37)(21,34)(22,38)(23,40)(26,31)(28,32)(35,39);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,25)(12,27)(14,31)(15,28)(16,24)(17,34)(18,36)(20,39)(21,33)(22,40)(23,38)(26,30)(29,32)(35,37);(9,14,16,15)(10,20,22,21)(11,26,27,32)(12,29,25,30)(13,31,24,28)(17,23,35,36)(18,34,38,37)(19,39,40,33)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=g,e^2=i,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, 6359231);