Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #9950 |
| commutator, Frattini, cosocle | C2×#131 | C23 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
330 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 60 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:
(1,2,8,13)(3,15,19,64)(4,22,27,81)(5,28,33,92)(6,35,39,79)(7,40,31,14)(9,42,47,17)(10,48,50,25)(11,52,56,34)(12,57,60,38)(16,66,23,82)(18,74,76,114)(20,53,21,65)(24,85,88,128)(26,90,41,59)(29,45,73,46)(30,80,95,117)(32,96,98,55)(36,63,77,120)(37,61,101,51)(43,103,108,125)(44,93,110,100)(49,111,112,94)(54,97,116,87)(58,102,109,72)(62,84,118,68)(67,123,83,78)(69,122,70,119)(71,105,86,121)(75,115,89,104)(91,113,126,99)(106,124,107,127);(1,3,7,21,8,19,31,20)(2,9,14,46,13,47,40,45)(4,23,26,69,27,16,41,70)(5,29,34,42,33,73,52,17)(6,36,38,18,39,77,57,76)(10,43,51,107,50,108,61,106)(11,53,28,64,56,65,92,15)(12,58,35,44,60,109,79,110)(22,78,59,105,81,123,90,121)(24,86,30,67,88,71,95,83)(25,89,37,84,48,75,101,68)(32,72,99,93,98,102,113,100)(49,62,54,115,112,118,116,104)(55,63,91,114,96,120,126,74)(66,117,122,128,82,80,119,85)(87,127,94,103,97,124,111,125);(1,4,8,27)(2,10,13,50)(3,16,19,23)(5,30,33,95)(6,37,39,101)(7,41,31,26)(9,43,47,108)(11,54,56,116)(12,59,60,90)(14,61,40,51)(15,62,64,118)(17,71,42,86)(18,75,76,89)(20,70,21,69)(22,79,81,35)(24,52,88,34)(25,57,48,38)(28,49,92,112)(29,67,73,83)(32,97,98,87)(36,68,77,84)(44,78,110,123)(45,107,46,106)(53,104,65,115)(55,117,96,80)(58,105,109,121)(63,119,120,122)(66,114,82,74)(72,124,102,127)(85,126,128,91)(93,103,100,125)(94,113,111,99)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=gi,e^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fgh,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,e^b=eg,f^b=fi,g^b=gi,d^c=dh,e^c=eg,g^c=gi,h^c=hi,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49899);