Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #124 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #13190 |
| prime radical | #575 | C8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
272 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 96 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:
(1,2,6,13,7,14,33,53,8,15,34,54,36,56,84,106,9,16,35,55,37,57,85,107,5,12,18,39,31,51,62,88)(3,17,22,58,23,59,70,108,20,41,60,52,68,92,109,105,21,49,32,81,69,102,83,119,29,10,38,42,79,43,86,93)(4,11,26,46,27,47,76,99,25,45,71,94,74,97,115,124,28,48,77,100,78,101,118,127,30,50,61,87,80,103,110,121)(19,63,66,98,67,111,114,126,24,40,72,90,73,91,116,123,65,44,82,95,113,96,120,125,64,89,75,104,112,122,117,128);(1,3,18,60,37,69,84,86,8,20,6,32,31,79,85,70,9,21,34,38,7,23,62,109,5,29,35,22,36,68,33,83)(2,10,39,58,57,92,106,119,15,17,13,52,51,102,107,93,16,41,54,81,14,43,88,108,12,49,55,42,56,59,53,105)(4,24,71,66,78,112,110,120,25,65,77,72,80,67,76,117,28,64,61,82,27,73,115,114,30,19,26,75,74,113,118,116)(11,44,94,90,101,111,121,128,45,89,100,95,103,91,99,126,48,63,87,104,47,96,124,123,50,40,46,98,97,122,127,125);(1,4)(2,11)(3,19,21,65)(5,30)(6,26)(7,27)(8,25)(9,28)(10,40,41,89)(12,50)(13,46)(14,47)(15,45)(16,48)(17,44,49,63)(18,61)(20,24,29,64)(22,66,32,82)(23,67,69,113)(31,80)(33,76)(34,71)(35,77)(36,74)(37,78)(38,75,60,72)(39,87)(42,90,52,104)(43,91,92,122)(51,103)(53,99)(54,94)(55,100)(56,97)(57,101)(58,95,81,98)(59,96,102,111)(62,110)(68,73,79,112)(70,114,83,120)(84,115)(85,118)(86,117,109,116)(88,121)(93,123,105,128)(106,124)(107,127)(108,125,119,126)Polycyclic presentation:
2^9|a^2=f,b^2=df,c^2=e,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 60081);