Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #153249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2234 |
| commutator, Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13539 |
| prime radical | C23 | #510 |
| LCS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
146 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 62 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 — 3 generators on 128 points:
(1,2)(3,17,20,61)(4,11,25,49)(5,12)(6,16)(7,13)(8,14)(9,15)(10,19,44,39)(18,63)(21,64,71,111)(22,58,72,119)(23,76)(24,46,40,62)(26,47,82,107)(27,48,83,109)(28,50)(29,51)(30,57)(31,60)(32,92,87,117)(33,53,88,114)(34,54)(35,55)(36,56)(37,96,90,43)(38,59,91,120)(41,69)(42,70,103,93)(45,80)(52,77,112,94)(65,113,81,97)(66,100)(67,99)(68,121)(73,115)(74,122)(75,98)(78,106)(79,105)(84,101)(85,102)(86,110,95,118)(89,116)(104,124)(108,126)(123,127)(125,128);(1,3,9,39,6,20,5,19)(2,10,12,17,16,44,15,61)(4,23,40,80,25,79,24,78)(7,32,36,94,30,87,28,77)(8,37,34,22,31,90,29,72)(11,45,46,76,49,106,62,105)(13,52,50,92,57,112,56,117)(14,58,51,96,60,119,54,43)(18,65,69,82,67,81,66,26)(21,73,93,89,71,84,70,35)(27,85,91,126,83,125,38,74)(33,75,95,124,88,123,86,68)(41,97,63,107,100,113,99,47)(42,101,64,116,103,115,111,55)(48,108,59,102,109,122,120,128)(53,104,110,98,114,121,118,127);(1,4,5,24,6,25,9,40)(2,11,12,46,16,49,15,62)(3,18,19,66,20,67,39,69)(7,33,28,86,30,88,36,95)(8,38,29,83,31,91,34,27)(10,41,17,63,44,100,61,99)(13,53,50,110,57,114,56,118)(14,59,51,109,60,120,54,48)(21,74,70,125,71,126,93,85)(22,75,37,68,72,123,90,124)(23,77,78,87,79,94,80,32)(26,84,81,89,82,73,65,35)(42,102,64,122,103,128,111,108)(43,104,58,98,96,121,119,127)(45,92,76,112,106,117,105,52)(47,101,97,116,107,115,113,55)Polycyclic presentation:
2^9|a^2=dh,b^2=de,c^2=d,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,f^b=fg,g^b=gh,h^b=hi,f^c=fg,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60614);