Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94210 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1384 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | #2108 | C22 |
| socle | C2 | #7218 |
| prime radical | C23 | #359 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C42 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ #2108 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
108 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 108 conjugacy classes of subgroups — too many to draw. Only its 46 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)(3,15,17,49)(4,21,23,66)(5,27)(6,31)(7,14)(8,34)(9,16)(10,25,40,72)(11,43)(12,22)(13,33)(18,61,58,111)(19,64,59,101)(20,65,60,52)(24,41,71,94)(26,76,73,68)(28,45)(29,79)(30,82)(32,39)(35,87)(36,91)(37,93)(38,56)(42,96,85,69)(44,81)(46,99)(47,70)(48,100)(50,104,102,97)(51,107,103,53)(54,77)(55,83)(57,84)(62,106,112,118)(63,115,113,105)(67,120,119,98)(74,116,121,78)(75,117,122,95)(80,86)(88,127)(89,108)(90,125)(92,128)(109,123)(110,124)(114,126);(1,3,5,18,22,53,43,97,33,60,81,113,6,19,28,62,7,17,29,58,70,107,99,104,8,20,30,63,32,59,80,112)(2,9,11,35,31,83,79,123,34,84,86,114,12,36,44,88,14,38,46,90,39,48,27,77,13,37,45,89,47,92,82,124)(4,16,24,54,69,91,98,87,73,93,122,127,25,55,74,108,23,56,71,109,96,128,120,125,26,57,75,110,72,100,121,126)(10,15,41,50,68,64,116,111,85,65,117,118,21,51,67,105,40,49,94,102,76,101,78,61,42,52,95,106,66,103,119,115)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fghi,e^2=fh,f^2=h,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gi,h^a=hi,d^b=dg,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1406);