Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #348082 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2847 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3531 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
747 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 747 conjugacy classes of subgroups — too many to draw. Only its 152 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:
(2,6)(4,11)(7,15)(8,12)(9,13)(14,16)(17,18,24,30)(19,31,36,47)(20,26,39,51)(21,40,42,54)(22,29,23,28)(25,35,48,34)(27,44,52,41)(32,55,59,72)(33,60,62,46)(37,65,67,69)(38,53,43,50)(45,63,71,58)(49,66,74,68)(56,64,73,61)(57,70,77,79)(75,76,80,78);(17,19,23,35,24,36,22,34)(18,25,29,47,30,48,28,31)(20,32,43,63,39,59,38,58)(21,33,44,64,42,62,41,61)(26,45,53,72,51,71,50,55)(27,46,54,73,52,60,40,56)(37,57,68,78,67,77,66,76)(49,70,65,80,74,79,69,75);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,20)(18,26)(19,32)(21,37)(22,38)(23,43)(24,39)(25,45)(27,49)(28,50)(29,53)(30,51)(31,55)(33,57)(34,58)(35,63)(36,59)(40,69)(41,66)(42,67)(44,68)(46,70)(47,72)(48,71)(52,74)(54,65)(56,75)(60,79)(61,76)(62,77)(64,78)(73,80);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,21)(18,27)(19,33)(20,37)(22,41)(23,44)(24,42)(25,46)(26,49)(28,40)(29,54)(30,52)(31,56)(32,57)(34,61)(35,64)(36,62)(38,66)(39,67)(43,68)(45,70)(47,73)(48,60)(50,69)(51,74)(53,65)(55,75)(58,76)(59,77)(63,78)(71,79)(72,80)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 255283);