Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6365322 |
| 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 | #62487 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,9)(3,11,14,27)(4,8,18,25)(5,19,22,10)(7,20,12,15)(13,29,31,24)(16,30,32,23)(17,28,21,26)(33,34)(35,40)(36,47)(37,42)(38,51)(39,43)(41,48)(44,59)(45,54)(46,56)(49,57)(50,55)(52,58)(53,61)(60,63)(62,64);(1,3)(2,7)(4,16)(5,20)(6,14)(8,24)(9,12)(10,27)(11,19)(13,17)(15,22)(18,32)(21,31)(23,26)(25,29)(28,30)(33,35,39,46)(34,40,43,56)(36,44,49,62)(37,50,38,45)(41,53,58,63)(42,55,51,54)(47,59,57,64)(48,61,52,60);(1,2)(3,12)(4,8)(5,10)(6,9)(7,14)(11,20)(13,30)(15,27)(16,29)(17,26)(18,25)(19,22)(21,28)(23,31)(24,32)(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)(54,59)(55,64)(56,63);(1,4,6,18)(2,8,9,25)(3,13,14,31)(5,21,22,17)(7,23,12,30)(10,28,19,26)(11,29,27,24)(15,32,20,16);(2,9)(3,14)(8,25)(10,19)(13,31)(15,20)(16,32)(26,28)(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=h,b^2=i,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,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=fh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272701);