Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #367 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1372 |
| prime radical | C23 | #368 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
132 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)(5,6,9,17,12,20,37,19,14,21,11,40,13,41,30,51)(7,22,24,52,26,54,56,68,28,55,36,44,27,63,61,53)(8,29,31,48,33,50,39,64,35,65,32,16,34,18,10,38)(15,42,43,59,46,62,67,23,47,57,49,58,66,25,45,60);(2,4)(5,7,10,25,11,24,35,60,12,26,31,42,30,56,34,59,14,28,39,62,9,36,8,23,13,27,32,57,37,61,33,58)(6,15,18,44,19,45,48,55,41,66,65,68,17,49,38,54,21,47,50,52,51,67,16,22,20,46,29,53,40,43,64,63)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=f,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efhi,f^a=fh,h^a=hi,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286896, #72286915, #72286916, … |
| maximal normal subgroup of | #72286896, #72286915, #72286916, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1224);