Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512832 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#3417 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
736 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 736 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) = 132 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 132 points:
(1,2)(3,4)(5,6,12,18)(7,19,24,61)(8,14,25,46)(9,29,34,58)(10,35,38,91)(11,39,37,17)(13,41,45,21)(15,50,53,32)(16,54,56,31)(20,59,62,111)(22,67,68,108)(23,69,40,60)(26,73,70,110)(27,78,71,119)(28,80,72,49)(30,55,86,109)(33,89,88,83)(36,51,92,107)(42,93,94,63)(43,90,97,85)(44,66,57,84)(47,100,98,76)(48,103,99,75)(52,87,82,106)(64,115,113,132)(65,116,81,112)(74,104,117,130)(77,126,118,122)(79,101,120,129)(95,127,128,124)(96,114,105,123)(102,125,121,131);(5,7,12,24)(6,13,18,45)(8,20,25,62)(9,30,34,86)(10,36,38,92)(11,40,37,23)(14,42,46,94)(15,51,53,107)(16,55,56,109)(17,57,39,44)(19,58,61,29)(21,50,41,32)(22,54,68,31)(26,74,70,117)(27,79,71,120)(28,81,72,65)(33,90,88,85)(35,43,91,97)(47,101,98,129)(48,104,99,130)(49,105,80,96)(52,108,82,67)(59,110,111,73)(60,83,69,89)(63,100,93,76)(64,103,113,75)(66,87,84,106)(77,127,118,124)(78,95,119,128)(102,132,121,115)(112,122,116,126)(114,125,123,131);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,25)(13,42)(15,47)(16,48)(17,49)(18,46)(19,59)(21,63)(22,64)(23,65)(24,62)(29,73)(30,74)(31,75)(32,76)(33,77)(34,70)(35,78)(36,79)(37,72)(38,71)(39,80)(40,81)(41,93)(43,95)(44,96)(45,94)(50,100)(51,101)(52,102)(53,98)(54,103)(55,104)(56,99)(57,105)(58,110)(60,112)(61,111)(66,114)(67,115)(68,113)(69,116)(82,121)(83,122)(84,123)(85,124)(86,117)(87,125)(88,118)(89,126)(90,127)(91,119)(92,120)(97,128)(106,131)(107,129)(108,132)(109,130);(8,25)(14,46)(20,62)(26,70)(27,71)(28,72)(42,94)(47,98)(48,99)(49,80)(59,111)(63,93)(64,113)(65,81)(73,110)(74,117)(75,103)(76,100)(77,118)(78,119)(79,120)(95,128)(96,105)(101,129)(102,121)(104,130)(112,116)(114,123)(115,132)(122,126)(124,127)(125,131)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420031);