Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #33982 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33984 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C16 | C25 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
1318 conjugacy classes of subgroups; only the 530 normal subgroups are shown.
This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 530 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 — 5 generators on 130 points:
(1,2);(3,4,10,16)(5,17,22,59)(6,12,23,44)(7,27,32,56)(8,33,36,89)(9,37,35,15)(11,39,43,19)(13,48,51,30)(14,52,54,29)(18,57,60,109)(20,65,66,106)(21,67,38,58)(24,71,68,108)(25,76,69,117)(26,78,70,47)(28,53,84,107)(31,87,86,81)(34,49,90,105)(40,91,92,61)(41,88,95,83)(42,64,55,82)(45,98,96,74)(46,101,97,73)(50,85,80,104)(62,113,111,130)(63,114,79,110)(72,102,115,128)(75,124,116,120)(77,99,118,127)(93,125,126,122)(94,112,103,121)(100,123,119,129);(3,5,10,22)(4,11,16,43)(6,18,23,60)(7,28,32,84)(8,34,36,90)(9,38,35,21)(12,40,44,92)(13,49,51,105)(14,53,54,107)(15,55,37,42)(17,56,59,27)(19,48,39,30)(20,52,66,29)(24,72,68,115)(25,77,69,118)(26,79,70,63)(31,88,86,83)(33,41,89,95)(45,99,96,127)(46,102,97,128)(47,103,78,94)(50,106,80,65)(57,108,109,71)(58,81,67,87)(61,98,91,74)(62,101,111,73)(64,85,82,104)(75,125,116,122)(76,93,117,126)(100,130,119,113)(110,120,114,124)(112,123,121,129);(3,6)(4,12)(5,18)(7,24)(8,25)(9,26)(10,23)(11,40)(13,45)(14,46)(15,47)(16,44)(17,57)(19,61)(20,62)(21,63)(22,60)(27,71)(28,72)(29,73)(30,74)(31,75)(32,68)(33,76)(34,77)(35,70)(36,69)(37,78)(38,79)(39,91)(41,93)(42,94)(43,92)(48,98)(49,99)(50,100)(51,96)(52,101)(53,102)(54,97)(55,103)(56,108)(58,110)(59,109)(64,112)(65,113)(66,111)(67,114)(80,119)(81,120)(82,121)(83,122)(84,115)(85,123)(86,116)(87,124)(88,125)(89,117)(90,118)(95,126)(104,129)(105,127)(106,130)(107,128);(6,23)(12,44)(18,60)(24,68)(25,69)(26,70)(40,92)(45,96)(46,97)(47,78)(57,109)(61,91)(62,111)(63,79)(71,108)(72,115)(73,101)(74,98)(75,116)(76,117)(77,118)(93,126)(94,103)(99,127)(100,119)(102,128)(110,114)(112,121)(113,130)(120,124)(122,125)(123,129)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532389);