Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #512952 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2215 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| prime radical | #33617 | 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 | |
413 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 136 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:
(1,2,9,16)(3,17,20,56)(4,25,29,52)(5,30,34,89)(6,15,8,13)(7,37,38,53)(10,39,41,21)(11,35,47,27)(12,48,51,18)(14,36,54,31)(19,65,66,110)(22,60,24,58)(23,76,70,111)(26,80,83,123)(28,84,85,46)(32,88,33,87)(40,94,95,119)(42,72,44,73)(43,100,97,113)(45,101,102,114)(49,63,50,64)(55,74,106,68)(57,93,99,61)(59,75,109,71)(62,78,103,104)(67,77,92,98)(69,120,117,108)(79,91,105,81)(82,86,90,122)(96,112,115,124)(107,125,126,128)(116,121,127,118);(1,3,18,61,7,23,32,72,6,22,63,100,36,75,34,39,9,20,48,93,38,70,33,73,8,24,64,113,31,71,5,21)(2,10,30,59,14,43,49,58,13,42,87,111,53,99,51,56,16,41,89,109,54,97,50,60,15,44,88,76,37,57,12,17)(4,19,62,112,28,69,90,94,35,74,101,125,91,121,83,92,29,66,103,124,85,117,82,119,27,68,114,128,81,118,26,67)(11,40,86,108,46,96,104,110,52,98,123,127,105,126,102,106,47,95,122,120,84,115,78,65,25,77,80,116,79,107,45,55);(1,4)(2,11)(3,19)(5,26)(6,35)(7,28)(8,27)(9,29)(10,40)(12,45)(13,52)(14,46)(15,25)(16,47)(17,55)(18,62)(20,66)(21,67)(22,74)(23,69)(24,68)(30,86)(31,81)(32,90)(33,82)(34,83)(36,91)(37,79)(38,85)(39,92)(41,95)(42,98)(43,96)(44,77)(48,103)(49,104)(50,78)(51,102)(53,105)(54,84)(56,106)(57,107)(58,110)(59,108)(60,65)(61,112)(63,101)(64,114)(70,117)(71,118)(72,94)(73,119)(75,121)(76,116)(80,88)(87,123)(89,122)(93,124)(97,115)(99,126)(100,125)(109,120)(111,127)(113,128);(3,20)(10,41)(17,56)(19,66)(21,39)(22,24)(23,70)(40,95)(42,44)(43,97)(55,106)(57,99)(58,60)(59,109)(61,93)(65,110)(67,92)(68,74)(69,117)(71,75)(72,73)(76,111)(77,98)(94,119)(96,115)(100,113)(107,126)(108,120)(112,124)(116,127)(118,121)(125,128)Polycyclic presentation:
2^9|a^2=i,b^2=eg,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,g^a=gh,h^a=hi,d^b=di,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420149);