Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6364762 |
| 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 | #19710 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1482 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)(3,11)(4,8)(5,18)(6,9)(7,13)(10,21)(12,29)(14,27)(15,32)(16,28)(17,25)(19,23)(20,26)(22,31)(24,30)(33,34)(35,40)(36,48)(37,42)(38,51)(39,43)(41,49)(44,58)(45,54)(46,55)(47,61)(50,56)(52,57)(53,59)(60,63)(62,64);(1,3,6,14)(2,7,9,23)(4,15,17,31)(5,19,21,13)(8,24,25,29)(10,27,18,11)(12,20,30,16)(22,28,32,26)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,53)(42,54)(43,55)(48,58)(49,59)(50,60)(51,61)(52,62)(56,63)(57,64);(1,2)(3,7)(4,8)(5,10)(6,9)(11,13)(12,22)(14,23)(15,24)(16,26)(17,25)(18,21)(19,27)(20,28)(29,31)(30,32)(33,36,39,50)(34,41,43,57)(35,44,46,60)(37,49,38,52)(40,53,55,64)(42,56,51,48)(45,59,47,62)(54,63,61,58);(1,4,6,17)(2,8,9,25)(3,12,14,30)(5,20,21,16)(7,22,23,32)(10,28,18,26)(11,29,27,24)(13,31,19,15)(33,37,39,38)(34,42,43,51)(35,45,46,47)(36,49,50,52)(40,54,55,61)(41,56,57,48)(44,59,60,62)(53,63,64,58);(2,9)(7,23)(8,25)(10,18)(11,27)(22,32)(24,29)(26,28)(35,46)(40,55)(44,60)(45,47)(53,64)(54,61)(58,63)(59,62)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^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=ei,f^b=fh,e^c=eh,g^c=gi,f^d=fh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6272062);