Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13528 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #508 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
196 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 196 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:
(1,2,5,13,34,59,103,122,9,15,38,61,45,70,41,66,10,17,6,14,35,60,79,71,7,42,36,65,8,16,37,62)(3,18,20,68,43,107,99,50,21,74,81,115,85,63,39,51,23,75,82,113,108,120,104,48,46,73,100,49,22,57,32,11)(4,24,26,87,44,109,102,67,30,88,94,55,98,58,40,12,31,53,27,69,92,121,101,54,28,96,93,119,29,64,33,56)(19,47,76,110,84,114,91,86,25,89,90,124,97,128,126,125,78,112,105,123,127,117,95,118,83,52,106,116,77,111,80,72);(1,3,7,21,10,23,9,46)(2,11,42,50,17,51,15,48)(4,25,28,78,31,83,30,19)(5,32,36,99,6,39,38,104)(8,43,35,85,45,108,34,22)(12,52,67,47,56,89,54,112)(13,57,65,107,14,63,61,120)(16,68,60,115,70,113,59,49)(18,71,74,66,75,122,73,62)(20,79,81,41,82,103,100,37)(24,86,96,125,53,118,88,72)(26,91,93,126,27,95,94,80)(29,97,92,127,98,77,44,84)(33,90,101,105,40,106,102,76)(55,111,87,114,119,128,69,117)(58,116,109,110,64,124,121,123);(1,4,10,31)(2,12,17,56)(3,19,23,78)(5,33,6,40)(7,30,9,28)(8,44,45,92)(11,47,51,112)(13,58,14,64)(15,67,42,54)(16,69,70,87)(18,72,75,125)(20,80,82,126)(21,83,46,25)(22,84,85,127)(24,62,53,66)(26,41,27,37)(29,35,98,34)(32,76,39,105)(36,102,38,101)(43,77,108,97)(48,89,50,52)(49,114,115,117)(55,60,119,59)(57,110,63,123)(61,109,65,121)(68,111,113,128)(71,96,122,88)(73,86,74,118)(79,94,103,93)(81,95,100,91)(90,99,106,104)(107,116,120,124)Polycyclic presentation:
2^9|a^2=d,b^2=degh,c^2=i,d^2=gh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dg,e^b=egi,g^b=gh,h^b=hi,d^c=dg,e^c=eg,f^c=fi,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60605);