Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6357861 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#127 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #60646 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,5)(3,7)(6,8)(9,10,15,19)(11,20,21,34)(12,17,26,33)(13,18,14,28)(16,23,29,24)(22,35,36,40)(25,31,27,32)(30,38,39,37);(9,11,14,24,15,21,13,23)(10,16,18,20,19,29,28,34)(12,22,25,37,26,36,27,38)(17,30,32,35,33,39,31,40);(1,2,3,5,4,6,7,8)(11,21)(16,29)(20,34)(22,36)(23,24)(30,39)(35,40)(37,38);(2,6)(5,8)(9,12)(10,17)(11,22)(13,27)(14,25)(15,26)(16,30)(18,32)(19,33)(20,35)(21,36)(23,38)(24,37)(28,31)(29,39)(34,40);(9,13,15,14)(10,18,19,28)(11,23,21,24)(12,25,26,27)(16,20,29,34)(17,31,33,32)(22,37,36,38)(30,40,39,35)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^c=di,e^d=eh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283125, #72283166, #72283179, … |
| maximal normal subgroup of | #72283125, #72283166, #72283179, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6265061);