Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94210 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1384 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C2×#504 | C22 |
| socle | C2 | #7230 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×#504 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
165 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 165 conjugacy classes of subgroups — too many to draw. Only its 56 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 — 2 generators on 128 points:
(1,2,5,11,8,14,31,15)(3,16,18,55,21,58,67,59)(4,22,24,73,28,75,27,84)(6,32,29,50,35,89,86,90)(7,36,30,13,33,49,23,53)(9,37,38,17,41,61,46,64)(10,42,43,26,45,76,47,80)(12,34,48,85,51,91,52,77)(19,68,65,107,71,118,117,119)(20,72,66,57,69,106,60,110)(25,79,78,104,83,44,82,101)(39,95,94,127,97,128,102,126)(40,98,54,93,99,63,103,111)(56,70,105,116,108,120,109,112)(62,114,113,96,115,87,92,125)(74,121,88,100,122,81,124,123);(1,3,6,19,33,69,91,120,5,18,29,65,23,60,77,112,8,21,35,71,7,20,34,70,31,67,86,117,30,66,85,116)(2,9,12,39,13,40,50,96,15,38,52,94,36,54,32,87,14,41,51,97,53,99,90,125,11,46,48,102,49,103,89,114)(4,17,25,62,80,98,121,128,27,37,82,92,76,111,123,127,28,64,83,115,26,63,81,95,24,61,78,113,42,93,100,126)(10,16,44,56,22,57,74,107,43,59,101,109,73,72,88,68,45,58,79,108,75,110,122,119,47,55,104,105,84,106,124,118)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fghi,d^2=i,e^2=fgi,f^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,h^a=hi,d^b=dg,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1416);