Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2087 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)(3,6)(4,5)(7,8)(9,10,15,21)(11,16,27,37)(12,13,18,31)(14,20,33,19)(17,32,30,29)(22,38,39,40)(23,35,34,28)(24,26,36,25);(2,5)(6,8)(9,11)(10,16)(12,28)(13,23)(14,25)(15,27)(17,38)(18,35)(19,36)(20,24)(21,37)(22,29)(26,33)(30,40)(31,34)(32,39);(9,12,15,18)(10,17,21,30)(11,22,27,39)(13,19,31,20)(14,32,33,29)(16,34,37,23)(24,38,36,40)(25,28,26,35);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,23)(12,21)(14,30)(15,31)(16,35)(17,33)(19,32)(20,29)(22,24)(25,40)(26,38)(27,34)(28,37)(36,39);(9,10)(11,24)(12,17)(13,29)(14,19)(15,21)(16,25)(18,30)(20,33)(22,38)(23,35)(26,37)(27,36)(28,34)(31,32)(39,40)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482650);