Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150163 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11678 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | #346 | C23 |
| prime radical | C23 | #361 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
189 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 189 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:
(1,2,6,13,8,15,40,61,9,17,7,41,26,16,43,14)(3,18,21,59,22,71,37,70,23,73,82,115,44,72,42,109)(4,24,28,55,29,57,98,66,30,56,38,107,75,11,51,54)(5,31,34,65,35,67,106,64,36,100,39,12,58,62,80,63)(10,32,46,20,48,83,69,85,50,102,116,86,49,84,47,108)(19,74,77,120,25,88,90,124,79,119,78,125,92,68,91,126)(27,93,95,87,97,99,117,60,33,103,96,122,105,121,127,53)(45,104,110,89,52,118,111,81,113,94,114,128,112,101,123,76);(1,3,9,23)(2,10,17,50)(4,25,30,92)(5,32,36,102)(6,37,7,42)(8,44,26,22)(11,52,57,112)(12,59,65,115)(13,47,41,69)(14,46,61,116)(15,49,16,48)(18,67,73,62)(19,75,79,29)(20,80,86,106)(21,43,82,40)(24,45,56,113)(27,94,33,104)(28,78,38,77)(31,72,100,71)(34,85,39,108)(35,84,58,83)(51,91,98,90)(53,120,60,125)(54,110,66,114)(55,123,107,111)(63,109,64,70)(68,121,88,99)(74,103,119,93)(76,96,81,95)(87,126,122,124)(89,117,128,127)(97,118,105,101);(1,4,26,75,9,30,8,29)(2,11,15,24,17,57,16,56)(3,19,44,92,23,79,22,25)(5,33,58,97,36,27,35,105)(6,38,43,98,7,28,40,51)(10,45,48,52,50,113,49,112)(12,60,63,122,65,53,64,87)(13,66,61,107,41,54,14,55)(18,68,71,74,73,88,72,119)(20,81,108,89,86,76,85,128)(21,78,42,90,82,77,37,91)(31,99,67,103,100,121,62,93)(32,101,84,94,102,118,83,104)(34,95,80,127,39,96,106,117)(46,114,69,123,116,110,47,111)(59,124,70,125,115,126,109,120)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=hi,d^2=fi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cg,d^b=dfh,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^c=ei,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57361);