Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153364 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13482 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| prime radical | C23 | #508 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
188 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(1,2,9,38,5,12,8,15)(3,16,36,44,18,10,21,59)(4,11,25,49,26,46,37,60)(6,13,39,95,27,50,33,56)(7,14,40,96,28,51,31,54)(17,61,66,105,67,41,74,100)(19,62,87,103,68,42,72,113)(20,63,84,57,69,43,34,94)(22,75,76,110,77,45,81,106)(23,47,78,108,64,98,89,115)(24,48,35,58,80,107,85,112)(29,86,92,114,82,52,88,118)(30,53,90,116,83,111,93,119)(32,55,97,128,70,101,91,117)(65,120,73,122,123,99,125,104)(71,121,79,127,124,102,126,109);(1,3,5,18)(2,10,12,16)(4,22,26,77)(6,29,27,82)(7,34,28,84)(8,36,9,21)(11,45,46,75)(13,52,50,86)(14,57,51,94)(15,59,38,44)(17,64,67,23)(19,70,68,32)(20,40,69,31)(24,79,80,126)(25,81,37,76)(30,73,83,125)(33,92,39,88)(35,71,85,124)(41,98,61,47)(42,101,62,55)(43,96,63,54)(48,109,107,127)(49,110,60,106)(53,104,111,122)(56,118,95,114)(58,102,112,121)(65,93,123,90)(66,78,74,89)(72,91,87,97)(99,119,120,116)(100,108,105,115)(103,117,113,128);(1,4)(2,11,12,46)(3,17)(5,26)(6,30)(7,35)(8,37)(9,25)(10,41,16,61)(13,53,50,111)(14,58,51,112)(15,60,38,49)(18,67)(19,71)(20,73)(21,74)(22,29)(23,32)(24,31)(27,83)(28,85)(33,93)(34,65)(36,66)(39,90)(40,80)(42,102,62,121)(43,104,63,122)(44,105,59,100)(45,86,75,52)(47,101,98,55)(48,96,107,54)(56,119,95,116)(57,99,94,120)(64,70)(68,124)(69,125)(72,126)(76,92)(77,82)(78,97)(79,87)(81,88)(84,123)(89,91)(103,127,113,109)(106,114,110,118)(108,117,115,128)Polycyclic presentation:
2^9|a^2=eh,b^2=i,c^2=d,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fg,g^b=gh,h^b=hi,e^c=ei,f^c=fg,g^c=gh,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60561);