Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | #120 |
| commutator | C23 | C22×C16 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12912 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
244 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 244 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:
(1,2,5,11,6,12,24,39,7,13,25,40,27,42,62,82,8,14,26,41,28,43,63,83,29,44,64,84,65,85,102,116)(3,9,17,32,18,33,53,73,19,34,54,74,55,75,97,111,16,31,50,70,51,71,94,108,52,72,95,109,96,110,122,126)(4,10,21,36,22,37,59,79,23,38,60,80,61,81,101,115,20,35,56,76,57,77,98,112,58,78,99,113,100,114,123,127)(15,45,47,86,48,87,91,117,49,88,92,118,93,119,121,128,46,30,66,67,89,68,103,105,90,69,104,106,120,107,124,125);(1,3,5,17,6,18,24,53,7,19,25,54,27,55,62,97,8,16,26,50,28,51,63,94,29,52,64,95,65,96,102,122)(2,9,11,32,12,33,39,73,13,34,40,74,42,75,82,111,14,31,41,70,43,71,83,108,44,72,84,109,85,110,116,126)(4,15,21,47,22,48,59,91,23,49,60,92,61,93,101,121,20,46,56,66,57,89,98,103,58,90,99,104,100,120,123,124)(10,30,36,67,37,68,79,105,38,69,80,106,81,107,115,125,35,45,76,86,77,87,112,117,78,88,113,118,114,119,127,128);(1,4)(2,10)(3,15)(5,21)(6,22)(7,23)(8,20)(9,30)(11,36)(12,37)(13,38)(14,35)(16,46)(17,47)(18,48)(19,49)(24,59)(25,60)(26,56)(27,61)(28,57)(29,58)(31,45)(32,67)(33,68)(34,69)(39,79)(40,80)(41,76)(42,81)(43,77)(44,78)(50,66)(51,89)(52,90)(53,91)(54,92)(55,93)(62,101)(63,98)(64,99)(65,100)(70,86)(71,87)(72,88)(73,105)(74,106)(75,107)(82,115)(83,112)(84,113)(85,114)(94,103)(95,104)(96,120)(97,121)(102,123)(108,117)(109,118)(110,119)(111,125)(116,127)(122,124)(126,128)Polycyclic presentation:
2^9|a^2=f,b^2=f,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59609);