Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #254539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
731 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)(3,17,20,10)(4,23,28,57)(5,12,32,8)(7,14,34,46)(9,16,31,44)(11,40,42,26)(15,36,33,45)(18,27,51,35)(19,21,54,38)(22,50,47,39)(24,29,55,60)(25,59,48,56)(30,49,62,37)(41,61,63,52)(43,64,58,53);(1,3)(2,10)(4,24,28,55)(5,19)(6,20)(7,33)(8,21)(9,22)(11,26,42,40)(12,38)(13,17)(14,36)(15,34)(16,39)(18,27,51,35)(23,29,57,60)(25,49,48,37)(30,59,62,56)(31,47)(32,54)(41,61,63,52)(43,53,58,64)(44,50)(45,46);(1,4,9,29)(2,11,16,43)(3,18,22,52)(5,30,33,25)(6,28,31,60)(7,26,19,53)(8,35,36,63)(10,37,39,56)(12,27,45,41)(13,42,44,58)(14,23,21,55)(15,48,32,62)(17,49,50,59)(20,51,47,61)(24,46,57,38)(34,40,54,64);(1,5,31,15)(2,12,44,36)(3,19,47,34)(4,25,60,62)(6,32,9,33)(7,20,54,22)(8,16,45,13)(10,38,50,14)(11,41,58,35)(17,21,39,46)(18,26,61,64)(23,56,24,49)(27,42,63,43)(28,48,29,30)(37,57,59,55)(40,52,53,51)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=i,d^2=hi,g^2=h,b^a=be,c^a=cf,d^a=deh,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,g^c=gh,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 161758);