Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235504 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1141 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13)(3,14,18,39)(4,19,23,10)(5,25,29,44)(7,12,28,43)(8,20,34,16)(9,26,38,17)(11,30,45,51)(15,46,49,42)(21,32,55,63)(22,35,58,60)(24,53,54,40)(27,47,48,36)(31,62,52,59)(33,57,41,61)(37,50,64,56);(1,3,6,18)(2,8,13,34)(4,20,23,16)(5,26,29,17)(7,15,28,49)(9,35,38,60)(10,39,19,14)(11,41,45,33)(12,32,43,63)(21,24,55,54)(22,50,58,56)(25,37,44,64)(27,51,48,30)(31,47,52,36)(40,42,53,46)(57,62,61,59);(1,3)(2,9)(4,21)(5,27)(6,18)(7,15)(8,31)(10,33)(11,42)(12,36)(13,38)(14,25)(16,24)(17,51)(19,41)(20,54)(22,56)(23,55)(26,30)(28,49)(29,48)(32,35)(34,52)(37,40)(39,44)(43,47)(45,46)(50,58)(53,64)(57,62)(59,61)(60,63);(3,15)(8,32)(9,36)(14,46)(16,21)(17,48)(18,49)(20,55)(26,27)(33,37)(34,63)(38,47)(39,42)(41,64)(50,57)(56,61)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=di,e^c=ei,f^c=fhi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142703);