Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512849 |
| 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 |
| 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 | |
732 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 732 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)(3,15)(4,22)(5,28)(6,34)(7,36)(8,11)(9,17)(10,24)(12,32)(13,35)(14,30)(16,58)(18,57)(19,69)(20,56)(21,55)(23,75)(25,46)(26,44)(27,47)(29,52)(31,92)(33,53)(37,50)(38,94)(39,96)(40,66)(41,65)(42,89)(43,64)(45,104)(48,63)(49,70)(51,110)(54,60)(59,72)(61,119)(62,113)(67,112)(68,108)(71,109)(73,79)(74,77)(76,115)(78,85)(80,111)(81,114)(82,95)(83,116)(84,103)(86,93)(87,91)(88,99)(90,101)(97,121)(98,124)(100,118)(102,125)(105,120)(106,117)(107,123)(122,128)(126,127);(1,3,8,21)(2,9,11,40)(4,16,27,62)(5,29,33,90)(6,18,35,20)(7,37,38,68)(10,39,44,97)(12,50,48,108)(13,41,34,43)(14,52,49,101)(15,53,55,28)(17,63,66,32)(19,70,71,30)(22,72,47,100)(23,76,81,124)(24,60,26,67)(25,83,84,122)(31,88,93,89)(36,42,94,99)(45,105,102,119)(46,107,103,127)(51,109,87,69)(54,111,112,85)(56,92,57,86)(58,114,113,75)(59,117,118,79)(61,82,120,77)(64,110,65,91)(73,116,106,128)(74,115,95,98)(78,123,80,126)(96,125,121,104);(1,4)(2,10)(3,16)(5,23)(6,26)(7,25)(8,27)(9,39)(11,44)(12,45)(13,47)(14,46)(15,54)(17,59)(18,67)(19,61)(20,60)(21,62)(22,34)(24,35)(28,85)(29,76)(30,77)(31,80)(32,79)(33,81)(36,95)(37,83)(38,84)(40,97)(41,100)(42,98)(43,72)(48,102)(49,103)(50,105)(51,106)(52,107)(53,111)(55,112)(56,113)(57,58)(63,117)(64,121)(65,96)(66,118)(68,122)(69,116)(70,82)(71,120)(73,87)(74,94)(75,92)(78,93)(86,114)(88,126)(89,123)(90,124)(91,125)(99,115)(101,127)(104,110)(108,119)(109,128);(2,11)(9,40)(10,44)(12,48)(13,34)(14,49)(15,55)(22,47)(28,53)(36,94)(39,97)(41,43)(42,99)(45,102)(46,103)(50,108)(51,87)(52,101)(54,112)(56,57)(58,113)(69,109)(72,100)(73,106)(74,95)(75,114)(85,111)(86,92)(98,115)(105,119)(107,127)(116,128)Polycyclic presentation:
2^9|b^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,g^a=gh,h^a=hi,e^b=eg,g^b=gh,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420046);