Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7548 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1424 |
| commutator | C32 | C2×C8 |
| Frattini, cosocle | C4×C32 | C22 |
| socle | C2 | #7461 |
| prime radical | C22 | #1425 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C8 ◁ C4×C16 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C4×C32 ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,12,7,14,8,41,9,17,35,15,40,16,42,58)(3,18,21,63,23,65,82,120,25,67,33,97,87,66,24,88)(4,26,28,89,30,91,31,96,32,93,94,92,95,44,70,108)(6,36,34,60,37,101,38,13,39,51,99,59,103,102,27,61)(10,43,45,105,47,78,109,20,49,75,57,77,48,114,104,79)(11,50,52,19,53,69,90,71,56,72,54,73,55,80,115,29)(22,83,81,121,84,126,74,122,86,117,98,107,128,127,85,64)(46,110,100,106,111,124,119,125,113,76,62,116,112,68,118,123);(1,3,19,68,6,22,70,114,35,21,80,123,99,81,32,79,7,23,71,110,37,84,28,43,42,82,50,106,27,74,95,105,9,25,73,124,39,86,31,78,5,33,69,125,34,98,4,20,40,87,29,76,103,128,94,75,8,24,72,116,38,85,30,77)(2,10,44,107,13,46,52,97,41,104,89,117,102,118,56,67,16,48,93,122,60,112,115,120,12,57,108,126,51,62,53,65,17,49,91,121,61,113,54,63,58,109,92,83,101,119,11,18,14,47,26,64,59,111,90,88,15,45,96,127,36,100,55,66)Polycyclic presentation:
2^9|a^2=d,b^2=ceh,c^2=egi,d^2=f,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,d^b=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72230739, #72230767, #72230778 |
| maximal normal subgroup of | #72230739, #72230767, #72230778 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1949);