Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141376 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9764 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | #405 | C23 |
| prime radical | C23 | #472 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ G | |
| LECS | G ▷ #405 ▷ C2×C4 ▷ C2 ▷ C1 | |
218 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 218 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,7,14,8,15,6,41,10,16,43,57,9,47,26,95)(3,17,21,69,22,70,85,68,23,71,42,110,46,62,44,111)(4,24,29,88,30,89,96,12,32,59,45,60,31,102,28,61)(5,33,37,104,38,65,75,49,40,105,18,72,39,13,36,64)(11,48,52,84,53,73,66,108,55,107,118,20,54,83,51,34)(19,76,80,67,25,90,93,123,81,127,79,114,94,124,92,120)(27,97,99,63,100,113,98,103,35,109,74,58,101,87,77,121)(50,82,86,125,56,106,115,126,117,78,119,128,116,91,122,112);(1,3,18,73,8,46,36,48,10,23,37,83,9,22,75,107)(2,11,49,110,47,53,104,111,16,55,64,69,15,54,72,68)(4,25,74,126,30,19,77,125,32,94,99,112,31,81,98,128)(5,34,7,44,38,20,6,42,40,108,43,85,39,84,26,21)(12,56,113,127,88,117,97,124,61,116,87,76,60,50,109,90)(13,62,57,51,105,17,41,52,65,70,14,66,33,71,95,118)(24,86,103,120,102,115,63,67,59,119,121,123,89,122,58,114)(27,82,29,80,100,91,96,92,35,78,45,79,101,106,28,93);(1,4,26,96,9,31,43,29,10,32,6,28,8,30,7,45)(2,12,57,59,15,60,95,102,16,61,14,24,47,88,41,89)(3,19,42,80,22,25,44,93,23,81,21,79,46,94,85,92)(5,35,75,98,39,100,37,99,40,27,36,77,38,101,18,74)(11,50,66,122,54,116,52,119,55,117,51,115,53,56,118,86)(13,63,64,87,33,103,104,97,65,58,49,113,105,121,72,109)(17,67,111,127,62,120,110,90,71,114,68,76,70,123,69,124)(20,82,83,128,34,106,48,112,84,78,73,125,108,91,107,126)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,c^2=ei,d^2=ghi,e^2=h,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,d^c=di,f^c=fi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48574);