Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535696 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3480 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,6,5)(3,8,4,7)(9,10,15,19,14,20,13,18)(11,21,23,31,25,34,24,16)(12,26,27,33,28,17,29,32)(22,36,38,40,39,35,37,30);(1,3,6,4)(2,7,5,8)(9,11,15,23,14,25,13,24)(10,16,18,34,20,31,19,21)(12,22,29,37,28,39,27,38)(17,30,32,36,26,40,33,35);(1,2,6,5)(3,8,4,7)(9,12,15,29,14,28,13,27)(10,17,18,32,20,26,19,33)(11,22,23,37,25,39,24,38)(16,30,34,36,31,40,21,35);(1,4,6,3)(2,7,5,8)(9,13,14,15)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,21,31,34)(17,32,26,33)(22,37,39,38)(30,35,40,36);(1,5,6,2)(3,8,4,7)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6461871);