Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1339 |
| commutator | C2×M32 | C2×C4 |
| Frattini, cosocle | #2113 | C22 |
| socle | C2 | #7077 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×M32 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ #2113 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
116 conjugacy classes of subgroups; only the 28 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 28 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:
(1,2,5,12,9,17,38,65)(3,18,20,71,25,74,84,126)(4,26,29,90,33,66,98,118)(6,39,42,105,28,14,35,62)(7,13,36,67,44,68,103,61)(8,15,37,63,43,16,69,64)(10,46,47,78,51,107,111,77)(11,52,54,30,58,99,60,41)(19,70,75,59,79,50,80,110)(21,85,87,128,45,53,81,116)(22,88,82,122,40,72,106,117)(23,73,83,125,24,89,34,102)(27,92,93,124,94,49,113,109)(31,57,96,121,101,56,115,120)(32,55,97,119,100,91,114,123)(48,95,112,86,104,127,108,76);(1,3,8,23,9,25,43,24)(2,10,15,50,17,51,16,70)(4,27,32,76,33,94,100,86)(5,34,37,20,38,83,69,84)(6,40,44,21,28,22,7,45)(11,53,56,72,58,85,57,88)(12,59,63,47,65,110,64,111)(13,49,39,104,68,92,14,48)(18,66,73,91,74,26,89,55)(19,41,77,101,79,30,78,31)(29,95,97,113,98,127,114,93)(35,87,36,82,42,81,103,106)(46,52,80,115,107,99,75,96)(54,117,120,128,60,122,121,116)(61,108,62,124,67,112,105,109)(71,119,125,118,126,123,102,90)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=fhi,d^2=i,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,h^a=hi,c^b=cf,d^b=defh,e^b=eh,f^b=fhi,h^b=hi,d^c=dghi,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 985);