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:
(1,2,5,11,7,35,30,13,8,14,31,49,36,15,23,50)(3,16,18,54,20,70,65,56,21,57,66,96,71,58,59,105)(4,22,24,10,27,43,74,92,28,73,26,45,77,48,82,46)(6,32,29,83,33,51,84,52,34,88,85,53,90,89,75,12)(9,37,38,93,91,95,101,17,41,60,47,62,42,102,40,63)(19,67,64,113,68,106,114,107,69,117,115,108,119,118,109,55)(25,78,76,87,80,104,122,72,81,44,79,103,123,120,124,121)(39,61,97,110,116,111,125,128,99,112,86,94,100,126,98,127);(1,3,6,19,31,66,85,115,7,20,33,68,23,59,75,109,8,21,34,69,5,18,29,64,36,71,90,119,30,65,84,114)(2,9,12,39,13,40,51,98,15,42,53,100,11,47,32,86,14,41,52,99,50,101,89,125,35,91,83,116,49,38,88,97)(4,17,25,61,24,60,76,110,27,62,80,111,74,102,122,128,28,63,81,112,26,37,79,94,77,93,123,126,82,95,124,127)(10,16,44,55,22,56,72,106,46,58,104,108,48,54,87,67,45,57,78,107,73,105,121,118,92,70,120,113,43,96,103,117)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fghi,d^2=gh,e^2=fgi,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, 1452);