Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2234 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13507 |
| prime radical | C23 | #510 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
154 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 66 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:
(1,2,5,27,6,13,9,12)(3,16,18,10,21,45,40,46)(4,11,23,51,26,52,41,48)(7,32,28,53,30,14,37,86)(8,15,29,87,31,55,35,54)(17,64,66,104,67,61,69,42)(19,71,70,108,74,62,98,43)(20,63,38,44,76,59,92,110)(22,77,79,47,80,117,81,112)(24,82,65,103,83,49,99,114)(25,50,39,60,85,116,93,113)(33,97,78,94,89,111,101,56)(34,95,88,118,90,57,102,119)(36,96,72,106,91,58,100,127)(68,121,122,120,123,109,75,105)(73,115,124,107,126,125,84,128);(1,3,5,18,6,21,9,40)(2,10,12,16,13,46,27,45)(4,22,23,79,26,80,41,81)(7,33,28,78,30,89,37,101)(8,38,29,76,31,92,35,20)(11,47,48,77,52,112,51,117)(14,56,53,111,32,94,86,97)(15,59,54,44,55,63,87,110)(17,65,66,83,67,99,69,24)(19,72,70,91,74,100,98,36)(25,84,39,73,85,124,93,126)(34,75,88,68,90,122,102,123)(42,103,61,82,104,114,64,49)(43,106,62,96,108,127,71,58)(50,115,113,128,116,125,60,107)(57,109,118,120,95,121,119,105);(1,4,5,23,6,26,9,41)(2,11,12,48,13,52,27,51)(3,17,18,66,21,67,40,69)(7,34,28,88,30,90,37,102)(8,39,29,85,31,93,35,25)(10,42,16,61,46,104,45,64)(14,57,53,118,32,95,86,119)(15,60,54,50,55,113,87,116)(19,73,70,124,74,126,98,84)(20,75,38,68,76,122,92,123)(22,78,79,89,80,101,81,33)(24,72,65,91,83,100,99,36)(43,107,62,115,108,128,71,125)(44,109,63,120,110,121,59,105)(47,111,77,94,112,97,117,56)(49,106,103,96,82,127,114,58)Polycyclic presentation:
2^9|a^2=eh,b^2=d,c^2=d,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fi,c^b=cf,e^b=ei,f^b=fg,g^b=gh,h^b=hi,e^c=ei,f^c=fg,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60583);