Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C16 | D8 |
| commutator | C8 | C2×C32 |
| Frattini, cosocle | C8×C16 | C22 |
| socle, prime radical | C22 | #1370 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C16 ◁ C4×C16 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
71 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 48 points:
(2,6)(3,7)(4,10)(8,16)(9,14)(11,15)(12,13)(17,18,19,23,20,24,27,33,21,25,28,34,30,36,39,43,22,26,29,35,31,37,40,44,32,38,41,45,42,46,47,48);(1,2,7,13,4,8,11,14,5,9,15,16,10,12,3,6)Polycyclic presentation:
2^9|a^2=d,b^2=ce,c^2=eg,d^2=f,e^2=g,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg176 conjugacy classes — view the full character table.
| maximal subgroup of | #72225400, #72225437, #72225443, … |
| maximal normal subgroup of | #72225400, #72225437, #72225443, … |
| maximal quotient of | #72225397 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1075);