Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573635 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60588 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1871 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,19)(13,20)(14,18)(15,21)(16,22)(23,32)(24,31)(25,33)(26,35)(27,36)(28,34)(29,38)(30,37)(39,40);(1,3,5,4)(2,6,8,7)(9,11,16,25)(10,17,22,33)(12,15,23,14)(13,27,29,26)(18,19,21,32)(20,36,38,35)(24,30,39,28)(31,37,40,34);(1,3,5,4)(2,7,8,6)(10,18)(11,23)(12,25)(14,15)(19,32)(20,34)(21,22)(24,27)(26,39)(28,30)(35,36)(37,38);(1,3,5,4)(2,7,8,6)(9,12,16,23)(10,19,22,32)(11,14,25,15)(13,24,29,39)(17,18,33,21)(20,31,38,40)(26,30,27,28)(34,35,37,36);(1,4,5,3)(2,6,8,7)(9,13,16,29)(10,20,22,38)(11,24,25,39)(12,26,23,27)(14,28,15,30)(17,31,33,40)(18,34,21,37)(19,35,32,36)Polycyclic presentation:
2^9|b^2=fi,c^2=f,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486237);