Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13700 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#13701 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2215 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 130 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 130 points:
(1,2);(3,4,11,18)(5,19,26,62)(6,27,31,54)(7,32,36,91)(8,17,10,15)(9,39,40,55)(12,41,46,22)(13,37,49,29)(14,50,53,20)(16,38,56,33)(21,67,71,111)(23,61,25,59)(24,77,78,112)(28,82,85,125)(30,86,87,48)(34,90,35,89)(42,96,98,120)(43,73,45,74)(44,101,102,115)(47,103,104,116)(51,65,52,66)(57,75,110,69)(58,95,100,63)(60,76,113,72)(64,80,105,106)(68,79,94,99)(70,121,122,109)(81,93,107,83)(84,88,92,124)(97,114,117,126)(108,127,128,130)(118,123,129,119);(3,5,20,63,9,24,34,73,8,23,65,101,38,76,36,41,11,26,50,95,40,78,35,74,10,25,66,115,33,72,7,22)(4,12,32,60,16,44,51,59,15,43,89,112,55,100,53,62,18,46,91,113,56,102,52,61,17,45,90,77,39,58,14,19)(6,21,64,114,30,70,92,96,37,75,103,127,93,123,85,94,31,71,105,126,87,122,84,120,29,69,116,130,83,119,28,68)(13,42,88,109,48,97,106,111,54,99,125,129,107,128,104,110,49,98,124,121,86,117,80,67,27,79,82,118,81,108,47,57);(3,6)(4,13)(5,21)(7,28)(8,37)(9,30)(10,29)(11,31)(12,42)(14,47)(15,54)(16,48)(17,27)(18,49)(19,57)(20,64)(22,68)(23,75)(24,70)(25,69)(26,71)(32,88)(33,83)(34,92)(35,84)(36,85)(38,93)(39,81)(40,87)(41,94)(43,99)(44,97)(45,79)(46,98)(50,105)(51,106)(52,80)(53,104)(55,107)(56,86)(58,108)(59,111)(60,109)(61,67)(62,110)(63,114)(65,103)(66,116)(72,119)(73,96)(74,120)(76,123)(77,118)(78,122)(82,90)(89,125)(91,124)(95,126)(100,128)(101,127)(102,117)(112,129)(113,121)(115,130)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420145);