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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,9)(3,11,14,21)(4,8,17,23)(5,18,19,10)(7,13,22,15)(12,25,27,31)(16,29,30,24)(20,26,32,28)(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);(1,3,6,14)(2,7,9,22)(4,12,17,27)(5,13,19,15)(8,20,23,32)(10,21,18,11)(16,26,30,28)(24,31,29,25)(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);(1,4,6,17)(2,8,9,23)(3,12,14,27)(5,16,19,30)(7,20,22,32)(10,24,18,29)(11,25,21,31)(13,26,15,28)(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);(3,13)(5,19)(7,21)(10,18)(11,22)(12,26)(14,15)(16,30)(20,31)(24,29)(25,32)(27,28);(2,9)(3,14)(8,23)(10,18)(12,27)(13,15)(24,29)(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=hi,b^2=h,c^2=hi,e^2=i,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, 6271869);