Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #121 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13120 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
207 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 207 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,6,13,7,14,32,48,8,12,33,38,35,47,80,85,9,15,34,49,36,50,81,96,5,30,17,55,31,79,57,103)(3,16,20,52,21,53,65,99,22,42,66,90,68,91,111,122,23,54,67,100,69,101,112,125,19,10,37,40,64,41,83,89)(4,24,27,72,28,73,76,116,29,71,77,102,78,115,117,126,26,11,43,45,75,46,93,95,25,44,56,84,74,94,104,119)(18,58,61,105,62,106,109,127,63,39,86,87,110,88,120,121,60,51,92,97,108,98,123,124,59,70,82,113,107,114,118,128);(1,3,17,37,36,69,80,111,8,22,6,20,31,64,81,112,9,23,33,66,7,21,57,83,5,19,34,67,35,68,32,65)(2,10,38,52,50,91,103,125,12,16,49,90,79,101,48,89,15,42,55,100,14,41,85,99,30,54,13,40,47,53,96,122)(4,18,56,82,75,108,117,120,29,63,27,61,74,107,93,123,26,60,77,86,28,62,104,118,25,59,43,92,78,110,76,109)(11,39,84,97,73,114,126,127,44,51,72,113,115,106,95,121,24,70,102,105,46,88,119,124,71,58,45,87,94,98,116,128);(1,4,5,25,9,26,8,29)(2,11,12,44,15,24,30,71)(3,18,19,59,23,60,22,63)(6,27,17,56,34,43,33,77)(7,28,31,74,36,75,35,78)(10,39,16,51,42,70,54,58)(13,45,38,84,49,72,55,102)(14,46,47,94,50,73,79,115)(20,61,37,82,67,92,66,86)(21,62,64,107,69,108,68,110)(32,76,57,104,81,93,80,117)(40,87,52,97,90,113,100,105)(41,88,53,98,91,114,101,106)(48,95,85,119,96,116,103,126)(65,109,83,118,112,123,111,120)(89,121,99,124,122,128,125,127)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=d,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^c=ei128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60005);