Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1402 |
| 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×D4 | C2×C16 |
| Frattini, cosocle | #2113 | C22 |
| socle | C2 | #7064 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C2×D4 ▷ 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:
(1,2,5,12,7,14,32,52,8,13,31,54,38,55,86,107,9,15,33,53,39,56,87,106,6,34,36,51,37,88,90,105)(3,16,18,57,20,59,66,109,22,60,65,108,71,111,115,128,21,61,67,93,70,112,116,124,19,58,29,81,69,110,82,121)(4,23,25,72,27,73,78,117,26,11,45,46,79,47,99,100,28,49,30,83,80,103,85,122,35,48,77,101,89,102,120,127)(10,40,41,91,43,92,95,17,42,62,50,64,97,113,104,63,44,84,96,114,98,123,126,24,74,75,94,76,118,119,125,68);(1,3,6,19,9,21,8,22)(2,10,13,42,15,44,34,74)(4,24,26,68,28,17,35,63)(5,29,31,18,33,65,36,67)(7,20,37,69,39,70,38,71)(11,16,23,58,48,61,49,60)(12,50,51,41,53,94,54,96)(14,43,55,97,56,98,88,118)(25,40,77,75,30,84,45,62)(27,76,79,91,80,64,89,114)(32,82,86,66,87,115,90,116)(46,81,83,57,101,108,72,93)(47,59,73,110,102,112,103,111)(52,104,105,95,106,125,107,126)(78,92,120,119,85,123,99,113)(100,121,122,109,127,128,117,124)Polycyclic presentation:
2^9|a^2=d,b^2=e,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=cg,d^b=dei,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, 930);