Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2184 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12919 |
| prime radical | C2×#41 | D8 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ C2×#504 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
252 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 252 conjugacy classes of subgroups — too many to draw. Only its 92 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:
(1,2,6,13,8,15,35,12)(3,16,19,60,22,61,21,59)(4,23,26,48,28,47,25,10)(5,29,32,57,34,88,93,87)(7,36,31,14,38,54,92,56)(9,39,42,69,44,68,41,18)(11,49,52,94,53,108,51,33)(17,62,65,111,67,112,64,58)(20,71,70,110,72,109,37,50)(24,76,80,113,82,121,79,86)(27,46,78,75,85,106,83,95)(30,55,89,101,91,100,90,43)(40,63,73,96,98,117,99,116)(45,81,74,107,104,126,103,125)(66,119,118,128,120,102,84,115)(77,114,123,127,124,97,122,105);(1,3,6,19,8,22,35,21)(2,9,13,42,15,44,12,41)(4,17,26,65,28,67,25,64)(5,30,32,89,34,91,93,90)(7,37,31,20,38,70,92,72)(10,40,23,73,48,98,47,99)(11,50,52,71,53,110,51,109)(14,55,54,101,56,100,36,43)(16,57,60,88,61,87,59,29)(18,49,39,94,69,108,68,33)(24,77,80,123,82,124,79,122)(27,84,78,66,85,118,83,120)(45,102,74,115,104,119,103,128)(46,105,75,114,106,127,95,97)(58,76,62,113,111,121,112,86)(63,107,96,126,117,125,116,81);(1,4)(2,10)(3,17)(5,24)(6,26)(7,27)(8,28)(9,40)(11,45)(12,47)(13,23)(14,46)(15,48)(16,58)(18,63)(19,65)(20,66)(21,64)(22,67)(25,35)(29,86)(30,77)(31,78)(32,80)(33,81)(34,82)(36,95)(37,84)(38,85)(39,96)(41,99)(42,73)(43,97)(44,98)(49,107)(50,102)(51,103)(52,74)(53,104)(54,75)(55,105)(56,106)(57,76)(59,112)(60,62)(61,111)(68,116)(69,117)(70,118)(71,115)(72,120)(79,93)(83,92)(87,121)(88,113)(89,123)(90,122)(91,124)(94,126)(100,127)(101,114)(108,125)(109,128)(110,119)Polycyclic presentation:
2^9|a^2=f,b^2=f,d^2=gh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,h^a=hi,d^b=dg,g^b=gh,h^b=hi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59621);