Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13669 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | C2×#13672 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#3417 | C2 |
| 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 | |
732 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 732 conjugacy classes of subgroups — too many to draw. Only its 146 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);(3,4,10,16)(5,17,23,61)(6,24,29,49)(7,30,35,89)(8,36,37,14)(9,38,40,96)(11,41,46,19)(12,28,50,26)(13,51,54,34)(15,55,57,32)(18,62,66,116)(20,60,22,59)(21,71,72,113)(25,76,82,124)(27,48,85,110)(31,56,92,103)(33,94,95,87)(39,52,70,114)(42,99,101,119)(43,67,45,68)(44,90,104,91)(47,105,108,127)(53,93,88,112)(58,69,115,64)(63,73,98,102)(65,120,121,106)(74,111,107,80)(75,83,97,78)(77,117,126,100)(79,86,81,123)(84,118,122,130)(109,128,129,125);(3,5)(4,11)(6,18)(7,31)(8,20)(9,39)(10,23)(12,42)(13,52)(14,43)(15,56)(16,46)(17,30)(19,34)(21,32)(22,37)(24,73)(25,77)(26,64)(27,84)(28,69)(29,66)(33,90)(35,92)(36,45)(38,44)(40,70)(41,51)(47,106)(48,109)(49,102)(50,101)(53,113)(54,114)(55,72)(57,103)(58,86)(59,87)(60,94)(61,89)(62,76)(63,80)(65,78)(67,93)(68,112)(71,88)(74,118)(75,117)(79,125)(81,128)(82,126)(83,121)(85,122)(91,95)(96,104)(97,100)(98,111)(99,105)(107,130)(108,120)(110,129)(115,123)(116,124)(119,127);(3,6,10,29)(4,12,16,50)(5,18,23,66)(7,25,35,82)(8,28,37,26)(9,27,40,85)(11,42,46,101)(13,47,54,108)(14,49,36,24)(15,48,57,110)(17,58,61,115)(19,63,41,98)(20,69,22,64)(21,65,72,121)(30,86,89,123)(31,77,92,126)(32,78,55,83)(33,81,95,79)(34,80,51,111)(38,97,96,75)(39,84,70,122)(43,102,45,73)(44,100,104,117)(52,106,114,120)(53,107,88,74)(56,109,103,129)(59,116,60,62)(67,99,68,119)(71,118,113,130)(76,87,124,94)(90,128,91,125)(93,105,112,127)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420043);