Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573652 |
| 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×#577 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,21)(11,23,25,17)(12,19,26,33)(13,20,29,37)(15,18,16,22)(24,39,28,32)(27,35,40,36)(30,34,31,38);(1,3,4,8)(2,5,6,7)(9,11,14,25)(10,17,21,23)(12,15,26,16)(13,27,29,40)(18,19,22,33)(20,36,37,35)(24,31,28,30)(32,38,39,34);(10,18)(11,12)(15,16)(19,33)(20,34)(21,22)(24,27)(25,26)(28,40)(30,31)(35,36)(37,38);(9,12)(10,19)(11,15)(13,28)(14,26)(16,25)(17,22)(18,23)(20,32)(21,33)(24,29)(27,30)(31,40)(34,35)(36,38)(37,39);(9,13)(10,20)(11,24)(12,27)(14,29)(15,30)(16,31)(17,32)(18,34)(19,35)(21,37)(22,38)(23,39)(25,28)(26,40)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,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, 6481774);