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 | |
2077 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)(11,17)(12,19)(13,27)(14,18)(15,21)(16,22)(20,30)(23,33)(24,39)(25,34)(26,37)(28,36)(29,38)(31,35)(32,40);(2,5)(6,8)(9,11,16,25)(10,17,22,34)(12,15,23,14)(13,28,30,26)(18,19,21,33)(20,37,27,36)(24,31,40,29)(32,38,39,35);(10,18)(11,23)(12,25)(14,15)(19,33)(20,35)(21,22)(24,28)(26,40)(27,38)(29,31)(36,37);(1,3,4,7)(2,6,5,8)(9,12)(10,19)(11,15)(13,24)(14,25)(16,23)(17,21)(18,34)(20,32)(22,33)(26,29)(27,39)(28,31)(30,40)(35,36)(37,38);(9,13)(10,20)(11,24)(12,26)(14,29)(15,31)(16,30)(17,32)(18,35)(19,36)(21,38)(22,27)(23,28)(25,40)(33,37)(34,39)Polycyclic presentation:
2^9|b^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, 6482646);