Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C128 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C4⋊C128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C64 | C22 |
| commutator | C2 | C2×C128 |
| socle, prime radical | C22 | C2×C64 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C2×C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
43 conjugacy classes of subgroups.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 2 generators on 132 points:
(2,4)(5,6,7,13,8,14,19,34,9,15,20,35,24,39,49,69,10,16,21,36,25,40,50,70,28,43,53,73,59,79,89,104,11,17,22,37,26,41,51,71,29,44,54,74,60,80,90,105,31,46,56,76,62,82,92,107,65,85,95,110,99,114,119,125,12,18,23,38,27,42,52,72,30,45,55,75,61,81,91,106,32,47,57,77,63,83,93,108,66,86,96,111,100,115,120,126,33,48,58,78,64,84,94,109,67,87,97,112,101,116,121,127,68,88,98,113,102,117,122,128,103,118,123,129,124,130,131,132);(1,2,3,4)Polycyclic presentation:
2^9|a^2=d,b^2=c,d^2=e,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bc320 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2034);