Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511969 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32930 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #585 | D4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3573 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
914 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 914 conjugacy classes of subgroups — too many to draw. Only its 184 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)(3,17,24,58)(4,25,29,52)(5,12,32,49)(6,33,35,50)(7,15,8,14)(10,38,43,21)(11,36,46,28)(13,34,51,18)(19,62,66,105)(20,55,69,104)(22,57,23,56)(26,76,79,74)(27,81,82,45)(30,83,84,100)(31,53,37,48)(39,89,91,107)(40,88,93,67)(41,70,42,59)(44,80,97,78)(47,86,98,61)(54,71,102,65)(60,75,85,99)(63,111,114,110)(64,73,87,94)(68,106,72,103)(77,96,122,123)(90,126,127,125)(92,108,95,116)(101,115,128,113)(109,120,121,118)(112,119,124,117);(1,3,18,59,8,23,35,38,9,24,34,70,7,22,6,21)(2,10,33,57,15,42,51,58,16,43,50,56,14,41,13,17)(4,19,60,107,28,65,82,87,29,66,85,89,36,71,27,64)(5,20,61,108,31,68,84,88,32,69,86,116,37,72,30,67)(11,39,75,62,25,73,81,102,46,91,99,105,52,94,45,54)(12,40,83,103,48,92,98,104,49,93,100,106,53,95,47,55)(26,63,109,126,80,115,122,124,79,114,121,125,78,113,77,112)(44,90,118,110,74,117,123,128,97,127,120,111,76,119,96,101);(1,4)(2,11)(3,19)(5,26)(6,27)(7,36)(8,28)(9,29)(10,39)(12,44)(13,45)(14,52)(15,25)(16,46)(17,54)(18,60)(20,63)(21,64)(22,71)(23,65)(24,66)(30,77)(31,80)(32,79)(33,75)(34,85)(35,82)(37,78)(38,87)(40,90)(41,94)(42,73)(43,91)(47,96)(48,74)(49,97)(50,99)(51,81)(53,76)(55,101)(56,105)(57,62)(58,102)(59,107)(61,109)(67,112)(68,115)(69,114)(70,89)(72,113)(83,118)(84,122)(86,121)(88,124)(92,117)(93,127)(95,119)(98,123)(100,120)(103,110)(104,128)(106,111)(108,126)(116,125);(1,5)(2,12)(3,20)(4,26)(6,30)(7,37)(8,31)(9,32)(10,40)(11,44)(13,47)(14,53)(15,48)(16,49)(17,55)(18,61)(19,63)(21,67)(22,72)(23,68)(24,69)(25,74)(27,77)(28,80)(29,79)(33,83)(34,86)(35,84)(36,78)(38,88)(39,90)(41,95)(42,92)(43,93)(45,96)(46,97)(50,100)(51,98)(52,76)(54,101)(56,106)(57,103)(58,104)(59,108)(60,109)(62,110)(64,112)(65,115)(66,114)(70,116)(71,113)(73,117)(75,118)(81,123)(82,122)(85,121)(87,124)(89,125)(91,127)(94,119)(99,120)(102,128)(105,111)(107,126)Polycyclic presentation:
2^9|a^2=i,b^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,h^a=hi,f^c=fi,g^c=gi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419166);