Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512941 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#2262 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
528 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 132 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 132 points:
(2,4)(5,6,12,18)(7,19,25,56)(8,14,26,43)(9,30,33,81)(10,34,36,83)(11,37,35,17)(13,38,42,22)(15,47,49,20)(16,50,51,31)(21,52,61,100)(23,67,69,116)(24,70,68,55)(27,74,71,117)(28,77,72,118)(29,78,73,46)(32,82,59,80)(39,85,89,62)(40,92,93,107)(41,66,84,88)(44,96,94,58)(45,98,95,75)(48,60,79,99)(53,87,104,57)(54,86,105,65)(63,114,110,132)(64,115,111,103)(76,121,108,120)(90,127,126,131)(91,113,122,125)(97,109,119,128)(101,124,129,106)(102,123,130,112);(5,7,20,57,10,23,59,88,35,68,99,92,50,86,33,38,12,25,47,87,36,69,32,66,11,24,60,107,31,65,9,22)(6,13,30,54,16,40,79,70,37,84,82,116,83,104,49,56,18,42,81,105,51,93,48,55,17,41,80,67,34,53,15,19)(8,21,58,106,28,63,108,125,73,111,128,127,98,123,71,85,26,61,96,124,72,110,76,113,29,64,109,131,75,112,27,62)(14,39,74,102,45,90,119,115,78,122,121,132,118,129,94,100,43,89,117,130,95,126,97,103,46,91,120,114,77,101,44,52);(1,2,3,4)(5,8)(6,14)(7,21)(9,27)(10,28)(11,29)(12,26)(13,39)(15,44)(16,45)(17,46)(18,43)(19,52)(20,58)(22,62)(23,63)(24,64)(25,61)(30,74)(31,75)(32,76)(33,71)(34,77)(35,73)(36,72)(37,78)(38,85)(40,90)(41,91)(42,89)(47,96)(48,97)(49,94)(50,98)(51,95)(53,101)(54,102)(55,103)(56,100)(57,106)(59,108)(60,109)(65,112)(66,113)(67,114)(68,111)(69,110)(70,115)(79,119)(80,120)(81,117)(82,121)(83,118)(84,122)(86,123)(87,124)(88,125)(92,127)(93,126)(99,128)(104,129)(105,130)(107,131)(116,132);(8,26)(14,43)(21,61)(27,71)(28,72)(29,73)(39,89)(44,94)(45,95)(46,78)(52,100)(58,96)(62,85)(63,110)(64,111)(74,117)(75,98)(76,108)(77,118)(90,126)(91,122)(97,119)(101,129)(102,130)(103,115)(106,124)(109,128)(112,123)(113,125)(114,132)(120,121)(127,131)Polycyclic presentation:
2^9|a^2=i,b^2=eg,c^2=f,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gh,h^a=hi,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420251);