Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #120 |
| commutator | C2×C4 | C22×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12976 |
| prime radical | #575 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
301 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 301 conjugacy classes of subgroups — too many to draw. Only its 108 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:
(1,2,6,13,7,14,32,51,8,15,33,52,35,54,81,102,9,16,34,53,36,55,82,103,5,12,29,48,30,49,78,100)(3,10,19,39,20,40,65,90,18,38,62,87,63,88,110,122,21,41,66,91,67,92,113,124,27,46,75,97,76,98,117,126)(4,22,25,69,26,70,74,115,24,43,72,50,73,95,116,101,23,47,31,79,71,99,80,118,28,11,42,44,77,45,94,96)(17,56,59,104,60,105,108,127,61,37,83,85,109,86,119,121,57,68,64,111,106,114,112,128,58,84,93,89,107,120,125,123);(1,3,6,19,7,20,32,65,8,18,33,62,35,63,81,110,9,21,34,66,36,67,82,113,5,27,29,75,30,76,78,117)(2,10,13,39,14,40,51,90,15,38,52,87,54,88,102,122,16,41,53,91,55,92,103,124,12,46,48,97,49,98,100,126)(4,17,31,64,26,60,80,112,24,61,42,93,73,109,94,125,23,57,25,59,71,106,74,108,28,58,72,83,77,107,116,119)(11,37,50,89,45,86,101,123,22,68,79,104,70,114,118,127,43,84,44,85,95,120,96,121,47,56,69,111,99,105,115,128);(1,4)(2,11)(3,17)(5,28)(6,31)(7,26)(8,24)(9,23)(10,37)(12,47)(13,50)(14,45)(15,22)(16,43)(18,61)(19,64)(20,60)(21,57)(25,34)(27,58)(29,72)(30,77)(32,80)(33,42)(35,73)(36,71)(38,68)(39,89)(40,86)(41,84)(44,53)(46,56)(48,69)(49,99)(51,101)(52,79)(54,70)(55,95)(59,66)(62,93)(63,109)(65,112)(67,106)(74,82)(75,83)(76,107)(78,116)(81,94)(85,91)(87,104)(88,114)(90,123)(92,120)(96,103)(97,111)(98,105)(100,115)(102,118)(108,113)(110,125)(117,119)(121,124)(122,127)(126,128)Polycyclic presentation:
2^9|a^2=f,b^2=df,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^b=ei,d^c=di,e^c=ei,f^c=fi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59685);