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)(18,26)(19,32)(21,38)(22,39)(23,46)(24,40)(25,47)(27,52)(28,53)(29,58)(30,54)(31,51)(33,61)(34,37)(35,62)(36,66)(41,69)(42,73)(43,70)(44,72)(45,71)(48,74)(49,67)(50,60)(55,78)(56,77)(57,76)(59,75)(63,80)(64,79)(65,68);(1,3)(2,8)(4,14)(5,12)(6,13)(7,11)(9,15)(10,16)(17,21,24,44)(18,27,30,56)(19,33,35,64)(20,38,40,72)(22,45,23,43)(25,48,49,63)(26,52,54,77)(28,57,29,41)(31,59,50,55)(32,61,62,79)(34,65,36,42)(37,68,66,73)(39,71,46,70)(47,74,67,80)(51,75,60,78)(53,76,58,69)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,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, 239222);