Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94210 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1384 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | #2108 | C22 |
| socle | C2 | #7242 |
| prime radical | C23 | #368 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ #2108 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,12,8,39,25,54,9,15,7,37,32,16,38,14)(3,17,19,58,22,76,63,109,23,61,21,74,69,62,75,60)(4,24,26,49,28,52,82,50,29,78,30,11,46,47,79,94)(6,33,31,57,35,55,80,13,36,53,34,91,87,90,92,56)(10,40,41,95,93,97,43,18,44,64,51,66,45,103,98,67)(20,70,68,112,72,110,113,59,73,108,71,119,116,118,120,111)(27,83,81,107,84,105,124,122,85,48,86,89,104,121,123,77)(42,99,88,96,117,126,100,127,101,65,106,114,102,115,125,128);(1,3,6,20,7,21,34,71,8,22,35,72,38,75,92,120,9,23,36,73,5,19,31,68,32,69,87,116,25,63,80,113)(2,10,13,42,54,98,90,125,16,45,57,102,12,51,53,106,15,44,56,101,14,43,55,100,39,93,91,117,37,41,33,88)(4,18,27,65,26,64,81,114,28,66,84,115,82,103,124,128,29,67,85,99,30,40,86,96,46,95,104,126,79,97,123,127)(11,17,48,59,78,109,122,118,50,62,105,112,52,58,107,108,49,61,83,111,24,60,77,110,94,76,121,119,47,74,89,70)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fghi,d^2=gh,e^2=fg,f^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,h^a=hi,d^b=dg,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1451);