Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514160 |
| 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 | C2×#454 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1666 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10,13,15)(11,19,22,33)(12,24,25,34)(14,28,27,18)(16,20,30,23)(17,29,31,26)(21,32,38,39)(35,37,40,36);(1,3,5,4)(2,6,8,7)(10,15)(11,20)(14,27)(16,19)(21,37)(22,23)(24,34)(26,29)(30,33)(32,35)(36,38)(39,40);(9,11)(10,16)(12,21)(13,22)(14,20)(15,30)(17,32)(18,33)(19,28)(23,27)(24,35)(25,38)(26,37)(29,36)(31,39)(34,40);(9,12,13,25)(10,17,15,31)(11,21,22,38)(14,29,27,26)(16,32,30,39)(18,34,28,24)(19,35,33,40)(20,36,23,37);(1,4,5,3)(2,7,8,6)(10,15)(16,30)(17,31)(18,28)(19,33)(24,34)(32,39)(35,40)Polycyclic presentation:
2^9|a^2=f,b^2=fi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426506);