Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #488723 |
| 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 | 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 | |
663 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 663 conjugacy classes of subgroups — too many to draw. Only its 136 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)(19,33,38,65)(20,39,43,62)(21,29,46,59)(22,28,45,60)(23,31,24,30)(26,50,54,36)(27,48,57,42)(34,66,71,77)(35,53,47,63)(37,52,49,61)(40,55,44,58)(41,75,76,56)(51,67,79,72)(64,70,80,69)(68,73,74,78);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19,25,38)(18,26,32,54)(20,34,43,71)(21,36,46,50)(22,47,45,35)(23,37,24,49)(27,51,57,79)(28,33,60,65)(29,61,59,52)(30,53,31,63)(39,73,62,78)(40,74,44,68)(41,67,76,72)(42,69,48,70)(55,80,58,64)(56,77,75,66);(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)(23,48)(24,42)(25,43)(26,51)(28,58)(29,56)(30,62)(31,39)(32,57)(33,64)(35,72)(36,68)(37,70)(38,71)(40,46)(45,76)(47,67)(49,69)(50,74)(52,66)(53,78)(54,79)(55,60)(59,75)(61,77)(63,73)(65,80);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21,24,45,25,46,23,22)(18,28,31,59,32,60,30,29)(19,35,37,36,38,47,49,50)(20,40,42,41,43,44,48,76)(26,52,53,33,54,61,63,65)(27,55,39,56,57,58,62,75)(34,67,70,74,71,72,69,68)(51,77,78,80,79,66,73,64)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=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, 396024);