Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153348 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13474 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #510 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
197 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 197 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:
(1,2,11,18,10,19,9,50)(3,20,52,79,27,80,26,95)(4,28,51,99,35,63,34,67)(5,36,53,15,40,71,39,16)(6,41,47,68,7,44,21,14)(8,48,54,70,43,107,38,17)(12,55,59,90,60,89,112,23)(13,37,65,31,66,106,75,32)(22,82,96,97,86,127,85,116)(24,91,111,126,45,58,81,77)(25,57,114,120,93,72,88,78)(29,100,113,122,103,76,87,61)(30,74,46,98,104,124,42,64)(33,73,109,123,105,69,49,62)(56,84,117,108,118,101,121,94)(83,115,110,125,102,128,92,119);(1,3,21,81,8,25,53,55,11,52,6,24,54,114,40,90,10,27,47,111,43,93,39,89,9,26,7,45,38,88,5,23)(2,12,36,78,17,58,44,95,50,112,16,72,107,126,68,80,19,60,71,120,70,91,41,79,18,59,15,57,48,77,14,20)(4,29,37,108,33,102,46,86,51,113,31,101,109,92,104,85,35,103,106,94,105,83,42,22,34,87,32,84,49,110,30,96)(13,61,67,82,64,115,69,56,75,76,63,116,124,119,123,121,66,122,99,127,98,128,73,118,65,100,28,97,74,125,62,117);(1,4,10,35)(2,13,19,66)(3,22,27,86)(5,37,40,106)(6,42,7,46)(8,49,43,109)(9,51,11,34)(12,56,60,118)(14,67,68,99)(15,69,16,73)(17,74,70,124)(18,75,50,65)(20,76,80,100)(21,30,47,104)(23,87,90,113)(24,92,45,110)(25,94,93,108)(26,96,52,85)(28,41,63,44)(29,55,103,89)(31,53,32,39)(33,54,105,38)(36,62,71,123)(48,64,107,98)(57,119,72,125)(58,82,91,127)(59,121,112,117)(61,79,122,95)(77,116,126,97)(78,115,120,128)(81,83,111,102)(84,114,101,88)Polycyclic presentation:
2^9|a^2=de,b^2=dg,c^2=i,d^2=gh,e^2=gi,f^2=gi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgh,g^a=gh,h^a=hi,c^b=cf,f^b=fi,d^c=dg,e^c=eghi,f^c=fghi,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, 60545);