Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1404 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | AΣL(1,𝔽16) |
| commutator | C2×Q8 | C2×C16 |
| Frattini, cosocle | #2113 | C22 |
| socle | C2 | #7064 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C2×Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2113 ◁ G | |
| LECS | G ▷ #2113 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
119 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 119 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,12,7,14,31,52,8,13,30,54,37,55,84,109,9,15,32,53,38,56,85,108,6,33,35,51,36,86,88,107)(3,16,18,57,20,59,66,110,22,60,28,80,72,112,81,121,21,45,67,92,71,99,115,124,19,58,69,100,70,111,117,127)(4,23,24,73,26,74,77,118,25,11,44,46,78,47,98,101,27,49,29,82,79,104,83,122,34,48,76,102,87,103,120,128)(10,39,40,89,42,90,93,65,41,91,94,114,96,123,125,68,43,75,95,116,97,119,126,17,61,62,50,64,105,113,106,63);(1,3,6,19,9,21,8,22)(2,10,13,41,15,43,33,61)(4,17,25,63,27,65,34,68)(5,28,30,67,32,69,35,18)(7,20,36,70,38,71,37,72)(11,45,23,60,48,16,49,58)(12,50,51,95,53,94,54,40)(14,42,55,96,56,97,86,105)(24,75,76,91,29,39,44,62)(26,64,78,89,79,114,87,116)(31,81,84,115,85,117,88,66)(46,100,82,57,102,80,73,92)(47,99,74,112,103,59,104,111)(52,106,107,126,108,125,109,93)(77,119,120,123,83,90,98,113)(101,127,122,110,128,121,118,124)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=f,e^2=i,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=cgi,d^b=de,g^b=gi,d^c=dgi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 931);