Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6366542 |
| 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 | |
1466 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)(3,13,18,24)(4,9,20,27)(5,11,6,10)(8,16,25,17)(14,29,32,23)(15,30,31,22)(19,28,21,26)(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)(2,8,12,25)(4,14,20,32)(5,17,6,16)(9,22,27,30)(10,13,11,24)(15,21,31,19)(23,28,29,26)(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)(5,21)(6,19)(7,20)(8,22)(10,28)(11,26)(12,27)(13,29)(15,16)(17,31)(18,32)(23,24)(25,30)(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)(5,19)(6,21)(7,20)(8,23)(10,26)(11,28)(12,27)(13,30)(14,16)(17,32)(18,31)(22,24)(25,29)(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)(4,19,20,21)(8,24,25,13)(9,26,27,28)(14,31,32,15)(22,29,30,23)(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,d^b=df,e^b=ehi,f^b=fh,e^c=eh,g^c=gi,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6273901);