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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1904 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)(3,6)(4,5)(7,8)(9,10)(11,19)(12,26)(13,31)(14,22)(15,21)(16,20)(17,25)(18,29)(23,35)(24,34)(27,33)(28,36)(30,38)(32,37)(39,40);(2,5)(6,8)(9,11,15,25)(10,17,21,19)(12,27,30,28)(13,14,24,16)(18,35,37,40)(20,34,22,31)(23,29,39,32)(26,36,38,33);(1,3,4,7)(2,6,5,8)(9,12,15,30)(10,18,21,37)(11,23,25,39)(13,28,24,27)(14,29,16,32)(17,33,19,36)(20,26,22,38)(31,40,34,35);(9,13)(10,19)(11,14)(12,28)(15,24)(16,25)(17,21)(18,36)(20,31)(22,34)(23,29)(26,40)(27,30)(32,39)(33,37)(35,38);(2,5)(6,8)(9,14,15,16)(10,20,21,22)(11,24,25,13)(12,29,30,32)(17,34,19,31)(18,26,37,38)(23,27,39,28)(33,35,36,40)Polycyclic presentation:
2^9|b^2=i,c^2=f,e^2=i,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, 7422092);