Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366710 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62548 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,18,25)(4,9,21,29)(5,11,6,10)(8,16,26,17)(14,24,19,32)(15,23,31,27)(20,30,22,28)(33,34,39,42)(35,40,46,54)(36,47,49,58)(37,43,38,51)(41,52,57,48)(44,59,62,64)(45,55,50,56)(53,60,63,61);(1,3,7,18)(2,8,12,26)(4,19,21,14)(5,17,6,16)(9,27,29,23)(10,13,11,25)(15,22,31,20)(24,30,32,28)(33,35,39,46)(34,40,42,54)(36,44,49,62)(37,50,38,45)(41,53,57,63)(43,56,51,55)(47,59,58,64)(48,61,52,60);(1,4)(2,9)(3,14)(5,22)(6,20)(7,21)(8,23)(10,30)(11,28)(12,29)(13,24)(15,17)(16,31)(18,19)(25,32)(26,27)(33,36)(34,41)(35,44)(37,48)(38,52)(39,49)(40,53)(42,57)(43,58)(45,60)(46,62)(47,51)(50,61)(54,63)(55,59)(56,64);(1,4)(2,9)(3,15)(5,20)(6,22)(7,21)(8,24)(10,28)(11,30)(12,29)(13,23)(14,17)(16,19)(18,31)(25,27)(26,32)(34,42)(40,54)(41,57)(43,51)(47,58)(53,63)(55,56)(59,64);(1,5,7,6)(2,10,12,11)(3,16,18,17)(4,20,21,22)(8,25,26,13)(9,28,29,30)(14,15,19,31)(23,24,27,32)(33,37,39,38)(34,43,42,51)(35,45,46,50)(36,48,49,52)(40,55,54,56)(41,58,57,47)(44,60,62,61)(53,59,63,64)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6274061);