Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7548 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1424 |
| commutator | C32 | C42 |
| Frattini, cosocle | C22×C32 | C22 |
| socle | C2 | #7471 |
| prime radical | C23 | Q64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C22×C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,12,9,17,34,16)(3,18,20,67,25,69,24,89)(4,13,27,55,30,62,92,61)(6,35,31,53,28,11,49,51)(7,39,32,98,36,50,96,103)(8,41,33,15,42,57,99,65)(10,43,44,19,48,71,54,74)(14,63,56,26,58,90,64,40)(21,78,75,124,72,66,82,121)(22,83,76,125,79,120,86,127)(23,87,77,122,88,128,84,68)(29,94,91,60,95,118,37,101)(38,102,97,100,93,59,114,52)(45,105,107,70,110,119,115,85)(46,111,108,123,109,126,116,81)(47,112,104,106,113,73,117,80);(1,3,6,21,36,79,91,106,33,84,97,126,90,105,27,43,5,24,31,82,96,76,95,112,42,88,93,123,40,85,30,74,9,25,28,72,7,22,37,80,99,77,114,111,63,119,92,71,34,20,49,75,32,86,29,73,8,23,38,81,26,70,4,19)(2,10,13,45,58,109,102,128,41,104,94,125,98,121,51,89,16,44,61,107,56,116,52,68,65,113,101,120,39,78,11,18,17,48,62,110,14,46,59,87,57,117,118,127,103,124,53,67,12,54,55,115,64,108,100,122,15,47,60,83,50,66,35,69)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgi,d^2=i,e^2=fi,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fg,g^a=gh,h^a=hi,d^b=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1962);