Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488723 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#447 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3259 | C22 |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 629 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 80 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2,6,4)(3,9,11,16)(5,13,15,8)(7,14,10,12)(17,18,25,32)(19,33,39,61)(20,40,44,65)(21,29,47,63)(22,48,46,28)(23,31,24,30)(26,52,57,36)(27,51,60,43)(34,68,72,77)(35,56,49,55)(37,54,38,67)(41,62,45,58)(42,64,50,59)(53,73,80,69)(66,75,78,71)(70,76,74,79);(1,3)(2,7)(4,10)(5,14)(6,11)(8,16)(9,13)(12,15)(17,19)(18,26)(20,34)(21,36)(22,49)(23,38)(24,37)(25,39)(27,53)(28,61)(29,54)(30,56)(31,55)(32,57)(33,48)(35,46)(40,76)(41,70)(42,73)(43,75)(44,72)(45,74)(47,52)(50,69)(51,71)(58,66)(59,77)(60,80)(62,78)(63,67)(64,68)(65,79);(1,4,6,2)(3,10,11,7)(5,8,15,13)(9,14,16,12)(17,20,25,44)(18,27,32,60)(19,34,39,72)(21,45,47,41)(22,50,46,42)(23,51,24,43)(26,53,57,80)(28,62,48,58)(29,64,63,59)(30,65,31,40)(33,66,61,78)(35,73,49,69)(36,74,52,70)(37,75,38,71)(54,68,67,77)(55,76,56,79);(3,11)(7,10)(9,16)(12,14)(17,21,24,46,25,47,23,22)(18,28,31,29,32,48,30,63)(19,35,38,36,39,49,37,52)(20,41,43,50,44,45,51,42)(26,54,56,61,57,67,55,33)(27,58,40,59,60,62,65,64)(34,69,71,70,72,73,75,74)(53,77,79,66,80,68,76,78)Polycyclic presentation:
2^9|a^2=h,c^2=h,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396060);