Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152839 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13149 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #510 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
199 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(1,2,7,15,10,17,9,49)(3,18,23,74,25,65,45,75)(4,26,30,93,32,69,46,95)(5,33,36,51,39,14,38,71)(6,40,44,108,37,13,19,67)(8,47,35,70,43,66,48,16)(11,50,54,21,57,84,56,34)(12,58,61,29,64,96,63,42)(20,77,80,117,82,52,27,59)(22,85,88,55,41,73,76,116)(24,90,89,68,87,112,83,53)(28,92,102,111,103,62,106,122)(31,60,107,94,105,110,101,123)(72,98,118,79,124,104,121,86)(78,115,100,91,109,113,97,119)(81,125,127,114,128,120,99,126);(1,3,19,76,8,24,39,21,7,45,6,41,35,83,38,50,10,25,44,88,43,87,5,34,9,23,37,22,48,89,36,84)(2,11,51,112,16,55,13,65,49,54,33,53,66,73,108,75,17,57,71,90,70,116,40,18,15,56,14,68,47,85,67,74)(4,27,96,79,31,99,28,100,46,20,29,104,101,81,106,109,32,80,58,86,105,127,103,97,30,82,42,98,107,128,102,78)(12,59,93,115,62,120,110,118,61,52,69,119,122,114,123,72,64,117,95,113,92,125,60,121,63,77,26,91,111,126,94,124);(1,4,5,28,35,101,19,58,9,30,38,102,8,31,37,29,10,32,39,103,48,107,44,96,7,46,36,106,43,105,6,42)(2,12,13,60,66,122,71,93,15,63,67,123,16,62,33,26,17,64,40,110,47,111,51,95,49,61,108,94,70,92,14,69)(3,20,21,78,83,127,88,104,23,27,84,97,24,81,41,79,25,82,34,109,89,99,76,98,45,80,50,100,87,128,22,86)(11,52,18,72,73,125,112,91,56,59,75,118,55,114,68,113,57,77,65,124,85,120,90,119,54,117,74,121,116,126,53,115)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=ghi,e^2=gi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60036);