Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347203 |
| 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 | #62752 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,14,19,32)(4,21,24,51)(5,26,28,11)(6,22,25,10)(8,13,31,39)(9,30,35,17)(15,36,44,34)(16,45,48,40)(18,41,20,42)(23,53,54,50)(27,37,55,52)(29,57,59,38)(33,46,56,58)(43,60,64,61)(47,63,49,62);(1,3,7,19)(2,9,12,35)(4,15,24,44)(5,16,28,48)(6,30,25,17)(8,20,31,18)(10,32,22,14)(11,33,26,56)(13,36,39,34)(21,41,51,42)(23,43,54,64)(27,58,55,46)(29,49,59,47)(37,45,52,40)(38,61,57,60)(50,62,53,63);(1,2)(3,9)(4,22)(5,11)(6,21)(7,12)(8,13)(10,24)(14,15)(16,33)(17,42)(18,34)(19,35)(20,36)(23,52)(25,51)(26,28)(27,50)(29,38)(30,41)(31,39)(32,44)(37,54)(40,43)(45,64)(46,63)(47,60)(48,56)(49,61)(53,55)(57,59)(58,62);(1,4,7,24)(2,10,12,22)(3,15,19,44)(5,23,28,54)(6,8,25,31)(9,32,35,14)(11,37,26,52)(13,21,39,51)(16,43,48,64)(17,18,30,20)(27,29,55,59)(33,45,56,40)(34,42,36,41)(38,50,57,53)(46,47,58,49)(60,63,61,62);(1,5,8,29)(2,11,13,38)(3,16,20,49)(4,23,25,55)(6,27,24,54)(7,28,31,59)(9,33,36,61)(10,37,21,50)(12,26,39,57)(14,40,42,63)(15,43,17,46)(18,47,19,48)(22,52,51,53)(30,58,44,64)(32,45,41,62)(34,60,35,56)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=h,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254432);