Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #246957 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #52059 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
999 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,15)(3,16,20,49)(4,21,22,40)(5,9,30,38)(6,31,35,13)(7,14,23,43)(10,25,41,19)(11,42,39,24)(12,45,47,26)(17,54,33,59)(18,56,27,62)(28,36,58,64)(29,53,60,37)(32,51,34,52)(44,55,50,61)(46,57,48,63);(2,9)(3,17)(4,22)(5,26)(6,32)(10,36)(11,40)(12,14)(13,48)(15,38)(16,50)(19,25)(20,33)(21,39)(24,58)(28,42)(29,60)(30,45)(31,46)(34,35)(37,52)(41,64)(43,47)(44,49)(51,53)(54,56)(57,63)(59,62);(1,3,7,17)(2,10,14,40)(4,6,25,34)(5,27,26,61)(8,20,23,33)(9,36,12,11)(13,49,51,59)(15,41,43,21)(16,52,54,31)(18,45,55,30)(19,32,22,35)(24,60,28,63)(29,58,57,42)(37,56,46,50)(38,64,47,39)(44,53,62,48);(1,4,23,19)(2,11,43,64)(3,6,33,32)(5,28,45,42)(7,25,8,22)(9,21,47,10)(12,41,38,40)(13,50,52,62)(14,36,15,39)(16,53,59,46)(17,34,20,35)(18,57,61,60)(24,30,58,26)(27,63,55,29)(31,44,51,56)(37,54,48,49)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fi,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 154178);