Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364250 |
| 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 | #62104 | 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) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,17,23)(4,9,20,27)(5,11,6,10)(8,15,24,16)(14,29,30,25)(18,32,31,22)(19,28,21,26)(33,34,39,43)(35,40,46,56)(36,47,49,57)(37,42,38,51)(41,52,58,48)(44,59,62,64)(45,54,50,55)(53,60,63,61);(1,3,7,17)(2,8,12,24)(4,18,20,31)(5,16,6,15)(9,25,27,29)(10,13,11,23)(14,21,30,19)(22,28,32,26)(33,35)(34,40)(36,44)(37,50)(38,45)(39,46)(41,53)(42,55)(43,56)(47,59)(48,61)(49,62)(51,54)(52,60)(57,64)(58,63);(33,36,39,49)(34,41,43,58)(35,44,46,62)(37,48,38,52)(40,53,56,63)(42,57,51,47)(45,60,50,61)(54,59,55,64);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,28)(12,27)(13,29)(15,18)(16,31)(17,30)(23,25)(24,32);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,20,21)(8,23,24,13)(9,26,27,28)(14,18,30,31)(22,25,32,29)(33,37,39,38)(34,42,43,51)(35,45,46,50)(36,48,49,52)(40,54,56,55)(41,57,58,47)(44,60,62,61)(53,59,63,64)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ehi,f^b=fh,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, 6271881);