Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #127 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13190 |
| prime radical | #575 | C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,5,11,6,12,25,41,7,13,26,42,28,44,67,89,8,14,27,43,29,45,68,90,30,46,69,91,70,92,110,122)(3,9,17,33,18,34,55,78,19,35,56,79,57,80,103,117,16,32,52,75,53,76,100,114,54,77,101,115,102,116,126,128)(4,10,22,38,23,39,65,87,24,40,48,72,66,88,97,112,21,37,59,82,63,85,106,119,64,86,96,93,109,121,125,123)(15,47,50,94,51,95,99,124,20,58,61,104,62,105,108,127,49,31,71,73,98,74,111,113,60,36,81,83,107,84,118,120);(1,3,5,17,6,18,25,55,7,19,26,56,28,57,67,103,8,16,27,52,29,53,68,100,30,54,69,101,70,102,110,126)(2,9,11,33,12,34,41,78,13,35,42,79,44,80,89,117,14,32,43,75,45,76,90,114,46,77,91,115,92,116,122,128)(4,20,59,81,23,62,106,118,24,49,96,50,66,98,125,99,21,60,22,61,63,107,65,108,64,15,48,71,109,51,97,111)(10,36,82,104,39,84,119,127,40,47,93,73,88,95,123,113,37,58,38,83,85,105,87,120,86,31,72,94,121,74,112,124);(1,4)(2,10)(3,15,16,49)(5,22)(6,23)(7,24)(8,21)(9,31,32,47)(11,38)(12,39)(13,40)(14,37)(17,50,52,71)(18,51,53,98)(19,20,54,60)(25,65)(26,48)(27,59)(28,66)(29,63)(30,64)(33,73,75,94)(34,74,76,95)(35,36,77,58)(41,87)(42,72)(43,82)(44,88)(45,85)(46,86)(55,99,100,111)(56,61,101,81)(57,62,102,107)(67,97)(68,106)(69,96)(70,109)(78,113,114,124)(79,83,115,104)(80,84,116,105)(89,112)(90,119)(91,93)(92,121)(103,108,126,118)(110,125)(117,120,128,127)(122,123)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ch,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60080);