Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7333 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #322 |
| commutator | C8 | C4×C16 |
| Frattini, cosocle | #1397 | C22 |
| socle | C2 | #7332 |
| prime radical | C22 | #1398 |
| LCS | G ▷ C8 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #1397 ◁ G | |
| LECS | G ▷ #1397 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
57 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,4,11,6,5,25,28,8,13,27,42,32,29,12,43,9,14,7,33,26,30,31,24,35,45,34,60,61,62,44,50)(3,15,16,46,18,40,51,63,21,49,19,55,52,41,57,64,22,17,53,54,20,48,38,39,58,56,23,47,59,10,36,37);(1,3,5,17,7,19,28,55,6,20,13,40,31,36,42,37,8,21,29,56,34,53,43,54,32,59,14,41,44,51,33,63,9,22,30,15,4,23,24,47,26,18,45,48,25,57,60,64,35,58,62,49,27,16,50,46,61,52,2,10,12,38,11,39)Polycyclic presentation:
2^9|a^2=c,b^2=d,c^2=e,d^2=f,e^2=h,f^2=g,g^2=h,h^2=i,b^a=be,d^a=dhi,f^a=fi,c^b=chi,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224644, #72224654, #72224659, … |
| maximal normal subgroup of | #72224644, #72224654, #72224659, … |
| maximal quotient of | #72230689, C2×#94509, C3×#94509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1705);