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 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,10,33)(3,16,37,62)(4,23,38,25)(5,28,39,32)(6,8,15,9)(7,24,40,46)(11,41,56,20)(12,27,29,26)(13,31,60,30)(14,35,58,36)(17,47,63,55)(18,53,64,52)(19,48,44,34)(21,45,22,49)(42,43,61,57)(50,51,54,59);(1,3)(2,11)(4,17)(5,18)(6,19)(7,34)(8,22)(9,21)(10,37)(12,42)(13,43)(14,45)(15,44)(16,24)(20,36)(23,54)(25,50)(26,53)(27,52)(28,59)(29,61)(30,55)(31,47)(32,51)(33,56)(35,41)(38,63)(39,64)(40,48)(46,62)(49,58)(57,60);(1,4,9,27)(2,12,6,25)(3,17,21,52)(5,24,31,14)(7,28,36,60)(8,26,10,38)(11,42,19,50)(13,40,32,35)(15,23,33,29)(16,47,45,18)(20,57,34,59)(22,53,37,63)(30,58,39,46)(41,43,48,51)(44,54,56,61)(49,64,62,55);(1,5)(2,13)(3,18)(4,24)(6,32)(7,29)(8,30)(9,31)(10,39)(11,43)(12,40)(14,27)(15,28)(16,17)(19,51)(20,54)(21,47)(22,55)(23,36)(25,35)(26,58)(33,60)(34,61)(37,64)(38,46)(41,50)(42,48)(44,59)(45,52)(49,53)(56,57)(62,63);(1,6,10,15)(2,8,33,9)(3,19,37,44)(4,25,38,23)(5,13,39,60)(7,24,40,46)(11,22,56,21)(12,26,29,27)(14,35,58,36)(16,48,62,34)(17,50,63,54)(18,43,64,57)(20,45,41,49)(28,31,32,30)(42,53,61,52)(47,51,55,59)Polycyclic presentation:
2^9|a^2=hi,c^2=h,e^2=hi,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, 6260105);