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:
(1,2,5,11,8,14,7,13)(3,15,17,34,21,47,20,55)(4,22,24,57,27,59,26,58)(6,29,28,43,31,42,23,12)(9,32,33,50,36,18,44,16)(10,37,38,54,40,60,39,25)(19,53,51,64,30,56,48,46)(35,49,45,61,41,63,62,52);(1,3,4,18,23,51,60,49,7,20,26,50,31,19,54,52,8,21,27,32,28,48,37,63,5,17,24,16,6,30,25,61)(2,9,10,34,29,62,57,46,11,33,38,47,43,35,59,53,14,36,40,55,42,45,58,64,13,44,39,15,12,41,22,56)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=i,e^2=fgi,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, 1558);