Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6349298 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | #62823 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,10)(3,11,13,24)(4,8,15,22)(5,17,19,9)(7,18,21,14)(12,25,26,32)(16,29,28,23)(20,30,31,27)(33,34)(35,39)(36,46)(37,49)(38,41)(40,48)(42,52)(43,59)(44,57)(45,54)(47,58)(50,55)(51,56)(53,61)(60,63)(62,64);(1,3)(2,7)(4,12)(5,18)(6,13)(8,20)(9,24)(10,21)(11,17)(14,19)(15,26)(16,30)(22,31)(23,32)(25,29)(27,28)(33,35,38,45)(34,39,41,54)(36,43,47,60)(37,50,52,44)(40,53,56,64)(42,57,49,55)(46,63,58,59)(48,62,51,61);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,25)(13,26)(14,27)(17,29)(18,30)(19,28)(21,31)(24,32)(33,36)(34,40)(35,43)(37,51)(38,47)(39,53)(41,56)(42,58)(44,62)(45,60)(46,49)(48,52)(50,61)(54,64)(55,63)(57,59);(4,15)(8,22)(12,26)(16,28)(20,31)(23,29)(25,32)(27,30)(34,41)(39,54)(40,56)(42,49)(46,58)(53,64)(55,57)(59,63);(3,13)(7,21)(11,24)(12,26)(14,18)(20,31)(25,32)(27,30)Polycyclic presentation:
2^9|a^2=h,b^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6256505);