Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150148 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11667 |
| 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 | |
217 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 217 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,6,13,8,15,39,59,9,17,7,40,26,16,42,14)(3,18,21,58,22,69,36,68,23,71,79,110,43,70,41,105)(4,24,28,54,29,56,97,64,30,55,37,103,73,11,50,53)(5,31,33,63,34,65,102,62,35,99,38,12,57,60,78,61)(10,32,45,20,47,80,67,82,49,101,111,83,48,81,46,104)(19,72,75,115,25,86,88,120,77,113,76,122,90,66,89,121)(27,91,93,85,95,125,112,117,96,118,94,116,119,98,128,52)(44,92,106,126,108,127,107,74,84,123,109,87,51,100,114,124);(1,3,9,23)(2,10,17,49)(4,25,30,90)(5,32,35,101)(6,36,7,41)(8,43,26,22)(11,51,56,108)(12,58,63,110)(13,46,40,67)(14,45,59,111)(15,48,16,47)(18,65,71,60)(19,73,77,29)(20,78,83,102)(21,42,79,39)(24,84,55,44)(27,92,96,123)(28,76,37,75)(31,70,99,69)(33,82,38,104)(34,81,57,80)(50,89,97,88)(52,115,117,122)(53,109,64,106)(54,107,103,114)(61,105,62,68)(66,98,86,125)(72,118,113,91)(74,94,124,93)(85,121,116,120)(87,112,126,128)(95,100,119,127);(1,4,26,73,9,30,8,29)(2,11,15,24,17,56,16,55)(3,19,43,90,23,77,22,25)(5,27,57,119,35,96,34,95)(6,37,42,97,7,28,39,50)(10,44,47,108,49,84,48,51)(12,52,61,85,63,117,62,116)(13,64,59,103,40,53,14,54)(18,66,69,72,71,86,70,113)(20,74,104,126,83,124,82,87)(21,76,41,88,79,75,36,89)(31,98,65,91,99,125,60,118)(32,100,81,123,101,127,80,92)(33,94,78,112,38,93,102,128)(45,109,67,114,111,106,46,107)(58,120,68,122,110,121,105,115)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=ch,d^b=dfh,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,e^d=ei50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57346);