Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6354981 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1514 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)(3,12)(4,9)(5,10)(6,20)(7,11)(8,16)(13,25)(14,22)(15,26)(17,23)(18,24)(19,30)(21,28)(27,31)(29,32)(33,34)(35,45)(36,50)(37,42)(38,43)(39,44)(40,49)(41,52)(46,60)(47,56)(48,55)(51,57)(53,58)(54,61)(59,62)(63,64);(1,3,6,16,7,15,5,14)(2,8,10,12,11,22,20,26)(4,13,19,28,18,29,17,27)(9,21,23,25,24,31,30,32)(33,35,37,49,39,47,38,48)(34,40,43,45,44,55,42,56)(36,46,52,62,53,61,51,63)(41,54,50,59,57,60,58,64);(1,4)(2,9)(3,13)(5,17)(6,19)(7,18)(8,21)(10,23)(11,24)(12,25)(14,27)(15,29)(16,28)(20,30)(22,31)(26,32)(33,36)(34,41)(35,46)(37,52)(38,51)(39,53)(40,54)(42,58)(43,50)(44,57)(45,59)(47,61)(48,63)(49,62)(55,60)(56,64);(1,5,7,6)(2,10,11,20)(3,14,15,16)(4,17,18,19)(8,12,22,26)(9,23,24,30)(13,27,29,28)(21,25,31,32)(35,47)(40,55)(45,56)(46,61)(48,49)(54,60)(59,64)(62,63);(3,15)(4,18)(8,22)(9,24)(12,26)(14,16)(17,19)(23,30)(33,37,39,38)(34,42,44,43)(35,48,47,49)(36,51,53,52)(40,45,55,56)(41,50,57,58)(46,62,61,63)(54,64,60,59)Polycyclic presentation:
2^9|b^2=f,d^2=hi,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,e^c=eh,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262235);