Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #490115 |
| 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 | C4×C8 | C24 |
| prime radical | #62341 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #2931 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
692 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 692 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,11)(4,9)(5,8)(6,10)(7,13)(12,15)(14,16)(17,18,24,30)(19,31,35,63)(20,26,40,56)(21,28,42,57)(22,27,41,58)(23,46,45,29)(25,48,51,34)(32,64,67,62)(33,52,43,61)(36,50,47,59)(37,60,70,54)(38,71,44,53)(39,73,72,55)(49,69,77,66)(65,79,75,78)(68,80,74,76);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,11)(10,16)(17,19,24,35)(18,25,30,51)(20,32,40,67)(21,34,42,48)(22,43,41,33)(23,47,45,36)(26,49,56,77)(27,31,58,63)(28,59,57,50)(29,61,46,52)(37,69,70,66)(38,65,44,75)(39,74,72,68)(53,62,71,64)(54,76,60,80)(55,79,73,78);(2,8)(4,5)(7,15)(9,10)(11,16)(13,14)(17,20)(18,26)(19,32)(21,37)(22,44)(23,39)(24,40)(25,49)(27,53)(28,60)(29,55)(30,56)(31,62)(33,65)(34,69)(35,67)(36,68)(38,41)(42,70)(43,75)(45,72)(46,73)(47,74)(48,66)(50,76)(51,77)(52,78)(54,57)(58,71)(59,80)(61,79)(63,64);(3,12)(7,16)(11,15)(13,14)(17,21,23,41,24,42,45,22)(18,27,29,57,30,58,46,28)(19,33,36,34,35,43,47,48)(20,37,39,38,40,70,72,44)(25,50,52,31,51,59,61,63)(26,53,55,54,56,71,73,60)(32,65,68,69,67,75,74,66)(49,76,78,62,77,80,79,64)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=ghi,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 397439);