Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574075 |
| 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 | |
1977 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)(3,8)(4,6)(5,7)(9,10,14,20)(11,22,24,17)(12,18,26,32)(13,19,27,36)(15,29,16,21)(23,35,39,31)(25,33,40,34)(28,38,30,37);(1,3,5,4)(2,6,7,8)(9,11,14,24)(10,17,20,22)(12,16,26,15)(13,25,27,40)(18,29,32,21)(19,34,36,33)(23,28,39,30)(31,38,35,37);(9,11,16,26,14,24,15,12)(10,18,29,17,20,32,21,22)(13,23,30,40,27,39,28,25)(19,33,38,31,36,34,37,35);(1,4,5,3)(2,6,7,8)(9,12,14,26)(10,18,20,32)(11,15,24,16)(13,23,27,39)(17,21,22,29)(19,35,36,31)(25,30,40,28)(33,37,34,38);(9,13)(10,19)(11,23)(12,25)(14,27)(15,28)(16,30)(17,31)(18,33)(20,36)(21,37)(22,35)(24,39)(26,40)(29,38)(32,34)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=f,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=dh,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, 6491541);