Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6352701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62909 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2604 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,22)(3,14,31,49)(4,7,13,8)(5,23,32,20)(6,21,33,39)(10,34,44,17)(11,24,51,26)(12,29,28,30)(15,40,36,27)(16,45,58,61)(18,38,19,42)(25,55,59,37)(35,56,63,62)(41,46,64,48)(43,47,52,60)(50,54,53,57);(1,3)(2,10)(4,15)(5,16)(6,27)(7,19)(8,18)(9,31)(11,35)(12,38)(13,36)(14,21)(17,30)(20,43)(22,44)(23,52)(24,46)(25,56)(26,48)(28,42)(29,34)(32,58)(33,40)(37,61)(39,49)(41,50)(45,55)(47,54)(51,63)(53,64)(57,60)(59,62);(1,4,8,2)(3,15,18,10)(5,20,24,11)(6,28,30,39)(7,22,9,13)(12,29,21,33)(14,40,38,34)(16,43,46,35)(17,49,27,42)(19,44,31,36)(23,26,51,32)(25,55,54,53)(37,57,50,59)(41,62,61,60)(45,47,64,56)(48,63,58,52);(1,5,7,26)(2,11,13,23)(3,16,19,48)(4,20,22,51)(6,25,29,57)(8,24,9,32)(10,35,36,52)(12,37,39,53)(14,41,42,45)(15,43,44,63)(17,47,40,62)(18,46,31,58)(21,50,28,55)(27,56,34,60)(30,54,33,59)(38,61,49,64);(5,24)(11,20)(16,46)(23,51)(25,54)(26,32)(35,43)(37,50)(41,61)(45,64)(47,56)(48,58)(52,63)(53,55)(57,59)(60,62)Polycyclic presentation:
2^9|a^2=hi,c^2=h,d^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=eh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260100);