Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7447 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1361 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C2 | #7274 |
| prime radical | C23 | #362 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
141 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 141 conjugacy classes of subgroups — too many to draw. Only its 76 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 64 points:
(1,2,5,11,8,14,7,32)(3,15,17,34,21,49,20,39)(4,22,24,58,27,44,26,60)(6,29,28,46,31,41,23,13)(9,18,33,53,37,52,36,16)(10,38,40,57,12,43,42,25)(19,56,54,64,30,62,50,48)(35,51,47,55,45,63,61,59);(1,3,4,18,23,54,43,63,7,20,26,16,31,19,57,55,8,21,27,52,28,50,38,51,5,17,24,53,6,30,25,59)(2,9,12,34,29,61,60,48,11,33,42,49,46,35,22,56,14,37,10,39,41,47,58,64,32,36,40,15,13,45,44,62)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=i,e^2=fg,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1557);