Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512841 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2184 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | #62635 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
738 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 738 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 128 points:
(1,2,8,14)(3,15,18,57)(4,10,21,43)(5,25,31,83)(6,32,35,95)(7,36,34,13)(9,38,40,17)(11,47,51,29)(12,52,54,28)(16,59,60,56)(19,67,66,111)(20,68,37,58)(22,71,27,80)(23,76,69,109)(24,78,70,46)(26,53,85,112)(30,93,92,82)(33,48,96,108)(39,64,97,61)(41,94,100,87)(42,65,55,86)(44,89,49,74)(45,88,101,73)(50,91,81,106)(62,118,116,121)(63,114,79,115)(72,104,84,113)(75,125,90,122)(77,102,119,107)(98,126,128,123)(99,117,105,120)(103,124,110,127);(1,3)(2,9)(4,16)(5,26)(6,33)(7,37)(8,18)(10,39)(11,48)(12,53)(13,55)(14,40)(15,25)(17,29)(19,28)(20,34)(21,60)(22,72)(23,77)(24,79)(27,84)(30,94)(31,85)(32,41)(35,96)(36,42)(38,47)(43,97)(44,102)(45,104)(46,105)(49,107)(50,111)(51,108)(52,66)(54,112)(56,80)(57,83)(58,93)(59,71)(61,74)(62,73)(63,70)(64,89)(65,106)(67,81)(68,82)(69,119)(75,126)(76,98)(78,99)(86,91)(87,92)(88,116)(90,123)(95,100)(101,113)(103,121)(109,128)(110,118)(114,122)(115,125)(117,127)(120,124);(1,4)(2,10)(3,16)(5,27)(6,23)(7,24)(8,21)(9,39)(11,49)(12,45)(13,46)(14,43)(15,56)(17,64)(18,60)(19,62)(20,63)(22,31)(25,80)(26,84)(28,73)(29,89)(30,90)(32,76)(33,77)(34,70)(35,69)(36,78)(37,79)(38,61)(40,97)(41,98)(42,99)(44,51)(47,74)(48,107)(50,110)(52,88)(53,104)(54,101)(55,105)(57,59)(58,114)(65,120)(66,116)(67,121)(68,115)(71,83)(72,85)(75,92)(81,103)(82,125)(86,117)(87,126)(91,127)(93,122)(94,123)(95,109)(96,119)(100,128)(102,108)(106,124)(111,118)(112,113);(4,21)(10,43)(16,60)(22,27)(23,69)(24,70)(39,97)(44,49)(45,101)(46,78)(56,59)(61,64)(62,116)(63,79)(71,80)(72,84)(73,88)(74,89)(75,90)(76,109)(77,119)(98,128)(99,105)(102,107)(103,110)(104,113)(114,115)(117,120)(118,121)(122,125)(123,126)(124,127)Polycyclic presentation:
2^9|a^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,f^b=fi,g^b=gh,h^b=hi,d^c=di,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420038);