Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #331999 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #532 | C23 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
499 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 499 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 128 |
A minimal faithful permutation representation — 4 generators on 80 points:
(1,2)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(17,18,24,30)(19,31,35,50)(20,26,40,54)(21,41,44,57)(22,29,23,28)(25,36,49,34)(27,45,56,43)(32,60,62,51)(33,63,64,48)(37,67,66,47)(38,69,72,76)(39,58,46,53)(42,55,65,59)(52,71,77,70)(61,74,79,80)(68,78,73,75);(1,2)(3,8)(4,9)(5,7)(6,10)(11,12)(13,16)(14,15)(17,19,24,35)(18,25,30,49)(20,37,40,66)(21,42,44,65)(22,36,23,34)(26,51,54,60)(27,55,56,59)(28,50,29,31)(32,46,62,39)(33,45,64,43)(38,61,72,79)(41,48,57,63)(47,58,67,53)(52,74,77,80)(68,71,73,70)(69,75,76,78);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,20,24,40)(18,26,30,54)(19,32,35,62)(21,38,44,72)(22,39,23,46)(25,47,49,67)(27,52,56,77)(28,53,29,58)(31,51,50,60)(33,61,64,79)(34,37,36,66)(41,69,57,76)(42,73,65,68)(43,70,45,71)(48,74,63,80)(55,78,59,75);(1,3,6,13)(2,8,10,16)(4,14,5,12)(7,11,9,15)(17,21)(18,27)(19,33)(20,38)(22,45)(23,43)(24,44)(25,48)(26,52)(28,57)(29,41)(30,56)(31,59)(32,61)(34,65)(35,64)(36,42)(37,68)(39,71)(40,72)(46,70)(47,74)(49,63)(50,55)(51,75)(53,76)(54,77)(58,69)(60,78)(62,79)(66,73)(67,80)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 239225);