Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488719 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
671 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 671 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:
(4,11)(6,12)(7,14)(9,15)(17,18,25,32)(19,33,39,67)(20,40,44,65)(21,29,47,61)(22,28,46,62)(23,31,24,30)(26,51,56,36)(27,50,59,43)(34,68,72,80)(35,55,48,54)(37,63,38,53)(41,57,45,60)(42,64,49,58)(52,73,79,69)(66,75,77,71)(70,76,74,78);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,26)(20,34)(21,36)(22,48)(23,38)(24,37)(25,39)(27,52)(28,33)(29,63)(30,55)(31,54)(32,56)(35,46)(40,76)(41,70)(42,73)(43,75)(44,72)(45,74)(47,51)(49,69)(50,71)(53,61)(57,77)(58,80)(59,79)(60,66)(62,67)(64,68)(65,78);(1,3,5,10)(2,6,8,12)(4,9,11,15)(7,13,14,16)(17,20,25,44)(18,27,32,59)(19,34,39,72)(21,45,47,41)(22,49,46,42)(23,50,24,43)(26,52,56,79)(28,60,62,57)(29,64,61,58)(30,65,31,40)(33,66,67,77)(35,73,48,69)(36,74,51,70)(37,75,38,71)(53,80,63,68)(54,76,55,78);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,24,46,25,47,23,22)(18,28,31,61,32,62,30,29)(19,35,38,36,39,48,37,51)(20,41,43,49,44,45,50,42)(26,53,55,33,56,63,54,67)(27,57,40,64,59,60,65,58)(34,69,71,70,72,73,75,74)(52,68,78,77,79,80,76,66)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=ghi,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dgh,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=fi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396020);