Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #141445 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9783 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×#131 | C23 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
317 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,15,20,44)(4,22,26,74)(5,28,32,86)(6,33)(7,13)(8,14)(9,18)(10,25,42,82)(11,16,47,59)(12,48)(17,60)(19,70,34,94)(21,55,36,96)(23,50)(24,77)(27,75,78,125)(29,87)(30,52)(31,89,91,46)(35,93)(37,97,100,121)(38,102)(39,68)(40,69)(41,104)(43,76,106,111)(45,109)(49,112)(51,56)(53,81)(54,95,108,61)(57,73)(58,103,71,85)(62,72,66,99)(63,98)(64,118)(65,119)(67,113)(79,84)(80,110)(83,123)(88,105)(90,92)(101,127,107,128)(114,126)(115,124,122,116)(117,120);(1,3,16,56,30,21,5,18,8,20,59,68,7,36,32,69)(2,9,28,55,13,51,47,15,14,40,86,96,52,39,11,44)(4,17,57,115,78,64,29,66,26,65,80,122,27,83,24,62)(6,34,58,102,48,95,31,90,35,19,71,112,23,61,91,67)(10,37,84,114,43,107,41,53,42,100,105,117,106,101,45,98)(12,49,85,70,33,92,89,108,50,38,103,94,93,113,46,54)(22,72,87,123,75,116,73,119,74,99,77,118,125,124,110,60)(25,81,109,127,111,126,88,121,82,63,104,128,76,120,79,97);(1,4,23,76,7,27,35,82,8,26,48,111,30,78,6,25)(2,10,33,75,52,106,12,22,14,42,93,125,13,43,50,74)(3,17,61,120,36,83,19,63,20,65,95,126,21,64,34,81)(5,29,58,109,16,57,91,79,32,24,71,104,59,80,31,88)(9,37,92,116,39,101,49,72,40,100,113,124,51,107,38,99)(11,45,85,87,86,105,46,110,47,41,103,77,28,84,89,73)(15,53,94,118,55,114,108,119,44,98,70,123,96,117,54,60)(18,66,102,127,56,115,67,97,69,62,112,128,68,122,90,121)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=gh,e^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48642);