Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #486327 |
| 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 | C23 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2960 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
643 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 643 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:
(6,12)(13,16)(17,18,25,32)(19,33,38,59)(20,39,43,62)(21,29,46,61)(22,47,45,28)(23,31,24,30)(26,51,55,36)(27,49,58,42)(34,65,70,79)(35,54,48,63)(37,72,50,53)(40,60,44,56)(41,74,76,57)(52,66,78,71)(64,69,80,68)(67,73,75,77);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19,25,38)(18,26,32,55)(20,34,43,70)(21,36,46,51)(22,48,45,35)(23,37,24,50)(27,52,58,78)(28,59,47,33)(29,53,61,72)(30,54,31,63)(39,73,62,77)(40,75,44,67)(41,66,76,71)(42,68,49,69)(56,64,60,80)(57,79,74,65);(1,3,5,10)(2,6,8,12)(4,9,11,15)(7,13,14,16)(17,20)(18,27)(19,34)(21,44)(22,41)(23,49)(24,42)(25,43)(26,52)(28,60)(29,57)(30,62)(31,39)(32,58)(33,64)(35,71)(36,67)(37,69)(38,70)(40,46)(45,76)(47,56)(48,66)(50,68)(51,75)(53,79)(54,77)(55,78)(59,80)(61,74)(63,73)(65,72);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,24,45,25,46,23,22)(18,28,31,29,32,47,30,61)(19,35,37,36,38,48,50,51)(20,40,42,41,43,44,49,76)(26,53,54,59,55,72,63,33)(27,56,39,74,58,60,62,57)(34,66,69,75,70,71,68,67)(52,65,77,64,78,79,73,80)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,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=fi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393979);