Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C64 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C64 | D4 |
| commutator | C4 | C2×C64 |
| Frattini, cosocle | C2×C64 | C22 |
| socle | C2 | C4⋊C64 |
| prime radical | C22 | M128 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C64 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C2×C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
41 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,13,6,14,28,48,7,15,29,49,33,52,74,95,8,16,30,50,34,53,75,96,37,55,78,98,81,100,114,123,9,12,31,44,35,45,76,89,38,46,79,90,82,92,115,119,23,47,64,91,66,93,107,120,67,94,108,121,109,122,126,128,10,17,32,51,36,54,77,97,39,56,80,99,83,101,116,124,19,11,40,41,60,42,84,86,61,43,85,87,105,88,117,118,4,24,25,68,26,69,71,110,27,70,72,111,73,112,113,127,3,18,20,57,21,58,62,102,22,59,63,103,65,104,106,125);(1,3,4,19,10,23,9,8)(2,11,12,18,17,16,24,47)(5,20,25,40,32,64,31,30)(6,21,26,60,36,66,35,34)(7,22,27,61,39,67,38,37)(13,41,44,57,51,50,68,91)(14,42,45,58,54,53,69,93)(15,43,46,59,56,55,70,94)(28,62,71,84,77,107,76,75)(29,63,72,85,80,108,79,78)(33,65,73,105,83,109,82,81)(48,86,89,102,97,96,110,120)(49,87,90,103,99,98,111,121)(52,88,92,104,101,100,112,122)(74,106,113,117,116,126,115,114)(95,118,119,125,124,123,127,128)Polycyclic presentation:
2^9|a^2=d,b^2=c,c^2=i,d^2=e,e^2=f,f^2=g,g^2=h,h^2=i,b^a=bc,c^a=ci224 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2035);