Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #317219 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3150 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
562 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 562 conjugacy classes of subgroups — too many to draw. Only its 126 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)(4,10)(5,9)(6,8)(7,13)(12,16)(14,15)(17,18,24,30)(19,31,35,49)(20,26,40,53)(21,41,44,54)(22,29,23,28)(25,36,48,34)(27,43,55,42)(32,58,60,50)(33,61,63,47)(37,66,65,46)(38,68,70,76)(39,56,45,52)(51,71,75,69)(57,64,73,62)(59,72,78,79)(67,77,80,74);(1,3)(2,7)(4,14)(5,13)(6,12)(8,15)(9,11)(10,16)(17,19,23,36,24,35,22,34)(18,25,29,49,30,48,28,31)(20,37,39,60,40,65,45,32)(21,33,43,64,44,63,42,62)(26,50,52,66,53,58,56,46)(27,47,54,73,55,61,41,57)(38,67,69,78,70,80,71,59)(51,74,68,79,75,77,76,72);(2,8)(4,6)(7,15)(9,10)(11,16)(12,14)(17,20)(18,26)(19,32)(21,38)(22,39)(23,45)(24,40)(25,46)(27,51)(28,52)(29,56)(30,53)(31,50)(33,59)(34,37)(35,60)(36,65)(41,68)(42,69)(43,71)(44,70)(47,72)(48,66)(49,58)(54,76)(55,75)(57,74)(61,79)(62,67)(63,78)(64,80)(73,77);(1,4,5,6)(2,8,9,10)(3,12,13,14)(7,16,11,15)(17,21,23,43,24,44,22,42)(18,27,29,54,30,55,28,41)(19,33,36,64,35,63,34,62)(20,38,45,71,40,70,39,69)(25,47,49,73,48,61,31,57)(26,51,56,76,53,75,52,68)(32,59,65,80,60,78,37,67)(46,72,58,77,66,79,50,74)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,f^b=fh,d^c=di,f^c=fi,g^c=gh50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 224417);