Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #145730 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10564 |
| commutator, Frattini, cosocle | C2×#131 | C23 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
298 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 58 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 — 3 generators on 128 points:
(1,2,9,16)(3,17,23,73)(4,24,31,65)(5,32,38,69)(6,39,42,14)(7,43,41,13)(8,46,34,15)(10,48,53,19)(11,40,59,27)(12,60,64,37)(18,74,79,117)(20,72,44,109)(21,71,84,122)(22,85,47,61)(25,89,83,121)(26,92,98,118)(28,57,45,58)(29,56,30,67)(33,68,105,52)(35,87,99,62)(36,63,100,102)(49,93,113,126)(50,90,66,81)(51,104,108,80)(54,103,115,127)(55,95,120,107)(70,82,124,76)(75,111,110,114)(77,125,78,116)(86,128,112,91)(88,97,123,96)(94,101,106,119);(1,3,9,23)(2,10,16,53)(4,25,31,83)(5,33,38,105)(6,21,42,84)(7,44,41,20)(8,47,34,22)(11,54,59,115)(12,61,64,85)(13,51,43,108)(14,66,39,50)(15,68,46,52)(17,69,73,32)(18,45,79,28)(19,60,48,37)(24,86,65,112)(26,93,98,126)(27,78,40,77)(29,76,30,82)(35,104,99,80)(36,90,100,81)(49,67,113,56)(55,117,120,74)(57,111,58,114)(62,122,87,71)(63,109,102,72)(70,123,124,88)(75,106,110,94)(89,118,121,92)(91,97,128,96)(95,103,107,127)(101,116,119,125);(1,4,5,26,34,30,60,97,9,31,38,98,8,29,37,96)(2,11,12,55,46,58,69,119,16,59,64,120,15,57,32,101)(3,18,19,75,47,78,105,127,23,79,48,110,22,77,33,103)(6,40,35,106,7,45,36,107,42,27,99,94,41,28,100,95)(10,49,17,70,68,86,85,89,53,113,73,124,52,112,61,121)(13,65,62,123,14,67,63,92,43,24,87,88,39,56,102,118)(20,82,80,93,84,25,90,91,44,76,104,126,21,83,81,128)(50,114,71,117,108,54,109,116,66,111,122,74,51,115,72,125)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=d,d^2=gi,e^2=gi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fhi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 52928);