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 | #62935 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2669 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14)(3,16,22,49)(4,23,27,11)(5,29,31,12)(6,13,35,28)(7,36,32,63)(9,15,24,44)(10,18,43,20)(17,47,53,56)(19,48,59,61)(21,40,50,62)(25,45,46,30)(26,34,38,33)(37,42,55,64)(39,51,58,60)(41,54,52,57);(1,3)(2,10)(4,17)(5,18)(6,19)(7,37)(8,22)(9,21)(11,39)(12,40)(13,41)(14,43)(15,42)(16,36)(20,31)(23,58)(24,50)(25,51)(26,61)(27,53)(28,52)(29,62)(30,54)(32,55)(33,56)(34,47)(35,59)(38,48)(44,64)(45,57)(46,60)(49,63);(1,4,24,13)(2,11,44,6)(3,17,50,41)(5,25,7,38)(8,27,9,28)(10,39,64,19)(12,45,63,33)(14,23,15,35)(16,47,62,54)(18,51,37,48)(20,60,55,61)(21,52,22,53)(26,31,46,32)(29,30,36,34)(40,57,49,56)(42,59,43,58);(1,5,9,32)(2,12,15,36)(3,18,21,55)(4,25,28,26)(6,33,23,30)(7,8,31,24)(10,40,42,16)(11,45,35,34)(13,38,27,46)(14,29,44,63)(17,51,52,61)(19,56,58,54)(20,50,37,22)(39,57,59,47)(41,48,53,60)(43,62,64,49);(1,6,24,11)(2,13,44,4)(3,19,50,39)(5,30,7,34)(8,35,9,23)(10,41,64,17)(12,46,63,26)(14,28,15,27)(16,48,62,51)(18,54,37,47)(20,57,55,56)(21,58,22,59)(25,36,38,29)(31,45,32,33)(40,60,49,61)(42,53,43,52)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^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, 6260076);