Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667760 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1766 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)(3,6)(4,8)(5,7)(9,10)(11,17)(12,25)(13,19)(14,29)(15,21)(16,22)(18,26)(20,32)(23,33)(24,27)(28,37)(30,40)(31,35)(34,39)(36,38);(1,3,6,2)(4,8,5,7)(9,11)(10,17)(12,22)(13,23)(14,30)(15,18)(16,25)(19,33)(20,36)(21,26)(24,28)(27,37)(29,40)(31,39)(32,38)(34,35);(9,12,16,17)(10,18,22,23)(11,21,25,19)(13,26,15,33)(14,27,32,34)(20,30,29,38)(24,37,39,35)(28,36,31,40);(1,4,6,5)(2,7,3,8)(9,13,16,15)(10,19,22,21)(11,23,25,18)(12,26,17,33)(14,28,32,31)(20,35,29,37)(24,38,39,30)(27,36,34,40);(1,5,6,4)(2,7,3,8)(9,14,16,32)(10,20,22,29)(11,24,25,39)(12,27,17,34)(13,28,15,31)(18,30,23,38)(19,35,21,37)(26,36,33,40)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584039);