Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7457 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1368 |
| commutator | C16 | C2×C16 |
| Frattini, cosocle | #1311 | C22 |
| socle | C2 | #7099 |
| prime radical | C22 | #1369 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C16 ◁ #1311 ◁ G | |
| LECS | G ▷ #1311 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
53 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,13,7,14,28,46,9,15,26,44,6,33,29,59,10,16,32,47,34,48,27,60,8,37,31,12,35,43,4,25)(3,17,19,39,21,50,55,62,23,45,30,61,57,64,53,11,24,38,51,40,36,49,18,41,22,58,52,63,20,56,54,42);(1,3,4,18,6,20,26,51,8,22,27,53,7,36,5,30,10,24,29,55,35,57,31,19,9,23,28,54,34,21,32,52)(2,11,12,17,14,39,25,49,15,41,13,45,33,61,46,56,16,42,44,38,48,40,59,50,37,62,47,58,43,63,60,64)Polycyclic presentation:
2^9|a^2=d,b^2=c,c^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,d^b=dgi,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224647, #72224648, #72224659, … |
| maximal normal subgroup of | #72224647, #72224648, #72224659, … |
| maximal quotient of | #72230695, C2×#93878, C3×#93878, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1074);