Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | D16 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1371 |
| prime radical | C23 | #368 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
136 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 68 points:
(1,2)(3,4)(5,6,9,17,12,20,30,19,14,21,11,38,13,39,35,52)(7,22,24,54,27,55,57,42,29,56,26,64,28,65,61,67)(8,18,31,48,32,50,37,49,34,51,10,16,33,46,36,47)(15,40,41,58,44,63,43,60,45,25,53,59,66,62,68,23);(2,4)(5,7,10,25,11,26,32,58,12,27,36,62,35,61,34,60,14,29,31,40,9,24,33,59,13,28,37,63,30,57,8,23)(6,15,18,42,19,43,49,65,39,66,46,54,17,41,48,56,21,45,51,67,52,68,47,55,20,44,50,64,38,53,16,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=g,e^2=f,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh,h^a=hi152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1223);