Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574086 |
| 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×#135 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1636 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)(11,16)(12,28)(13,18)(14,20)(15,21)(17,29)(19,32)(22,39)(23,34)(24,26)(25,38)(27,35)(30,33)(31,37)(36,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,23)(13,31)(14,25)(15,27)(17,36)(18,37)(19,24)(20,38)(21,35)(22,30)(26,32)(28,34)(29,40)(33,39);(9,12,14,29,15,13,32,30)(10,17,19,28,21,33,20,18)(11,22,25,23,27,40,26,31)(16,34,24,39,35,37,38,36);(9,13,15,12)(10,18,21,28)(11,23,27,31)(14,30,32,29)(16,34,35,37)(17,20,33,19)(22,25,40,26)(24,39,38,36);(9,10)(11,24)(12,17)(13,33)(14,19)(15,21)(16,26)(18,30)(20,32)(22,39)(23,37)(25,35)(27,38)(28,29)(31,34)(36,40)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,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, 6490563);