Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #513188 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | D4×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #443 ◁ #2170 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
414 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 118 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:
(1,2,10,18)(3,19,25,79)(4,26,31,70)(5,32,37,105)(6,38,44,96)(7,17,9,15)(8,47,50,71)(11,53,58,22)(12,45,60,29)(13,61,28,49)(14,66,27,42)(16,46,73,40)(20,80,86,119)(21,62,90,97)(23,92,51,78)(24,93,94,125)(30,100,103,59)(33,55,108,84)(34,63,109,98)(35,65,52,99)(36,64,110,75)(39,72,114,126)(41,111,43,102)(48,67,117,95)(54,77,121,128)(56,91,74,113)(57,116,81,112)(68,115,69,104)(76,85,127,83)(82,88,118,120)(87,124,89,123)(101,106,122,107);(1,3)(2,11)(4,20)(5,33)(6,39)(7,23)(8,48)(9,51)(10,25)(12,54)(13,62)(14,67)(15,56)(16,72)(17,74)(18,58)(19,38)(21,34)(22,42)(24,40)(26,88)(27,95)(28,97)(29,83)(30,101)(31,86)(32,77)(35,106)(36,76)(37,108)(41,112)(43,116)(44,114)(45,85)(46,94)(47,57)(49,80)(50,117)(52,107)(53,66)(55,63)(59,89)(60,121)(61,119)(64,123)(65,118)(68,93)(69,125)(70,120)(71,81)(73,126)(75,124)(78,102)(79,96)(82,99)(84,98)(87,100)(90,109)(91,115)(92,111)(103,122)(104,113)(105,128)(110,127);(1,4,10,31)(2,12,18,60)(3,20,25,86)(5,27,37,14)(6,28,44,13)(7,45,9,29)(8,30,50,103)(11,54,58,121)(15,70,17,26)(16,59,73,100)(19,76,79,127)(21,81,90,57)(22,82,53,118)(23,85,51,83)(24,84,94,55)(32,104,105,115)(33,95,108,67)(34,71,109,47)(35,68,52,69)(36,96,110,38)(39,97,114,62)(40,98,46,63)(41,64,43,75)(42,99,66,65)(48,101,117,122)(49,102,61,111)(56,120,74,88)(72,89,126,87)(77,113,128,91)(78,119,92,80)(93,107,125,106)(112,123,116,124);(1,5)(2,13)(3,21)(4,27)(6,12)(7,35)(8,49)(9,52)(10,37)(11,55)(14,31)(15,64)(16,32)(17,75)(18,28)(19,77)(20,81)(22,80)(23,87)(24,54)(25,90)(26,41)(29,69)(30,102)(33,39)(34,42)(36,40)(38,63)(43,70)(44,60)(45,68)(46,110)(47,65)(48,88)(50,61)(51,89)(53,119)(56,101)(57,86)(58,84)(59,104)(62,67)(66,109)(71,99)(72,85)(73,105)(74,122)(76,113)(78,82)(79,128)(83,126)(91,127)(92,118)(93,123)(94,121)(95,97)(96,98)(100,115)(103,111)(106,112)(107,116)(108,114)(117,120)(124,125)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fi,g^a=gh,h^a=hi,d^b=de,e^b=eg,g^b=gh,h^b=hi,f^c=fi,e^d=eg,g^d=gh,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420388);