Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6355044 |
| 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 |
| prime radical | #60961 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1491 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)(3,11,13,25)(4,8,15,22)(5,17,18,9)(7,19,21,14)(12,24,26,32)(16,29,28,23)(20,30,31,27)(33,34)(35,40)(36,48)(37,42)(38,51)(39,43)(41,49)(44,59)(45,54)(46,55)(47,56)(50,57)(52,58)(53,61)(60,64)(62,63);(1,3,5,14,6,13,18,19)(2,7,9,11,10,21,17,25)(4,12,16,27,15,26,28,30)(8,20,23,24,22,31,29,32)(33,35)(34,40)(36,44)(37,46)(38,45)(39,47)(41,53)(42,55)(43,56)(48,59)(49,61)(50,62)(51,54)(52,60)(57,63)(58,64);(1,4)(2,8)(3,12)(5,16)(6,15)(7,20)(9,23)(10,22)(11,24)(13,26)(14,27)(17,29)(18,28)(19,30)(21,31)(25,32)(33,36)(34,41)(35,44)(37,49)(38,52)(39,50)(40,53)(42,57)(43,58)(45,60)(46,61)(47,62)(48,51)(54,59)(55,63)(56,64);(3,13)(7,21)(11,25)(12,26)(14,19)(20,31)(24,32)(27,30)(33,37,39,38)(34,42,43,51)(35,45,47,46)(36,49,50,52)(40,54,56,55)(41,57,58,48)(44,60,62,61)(53,59,64,63);(3,13)(4,15)(7,21)(8,22)(11,25)(14,19)(16,28)(23,29)(33,37,39,38)(34,42,43,51)(35,46,47,45)(36,49,50,52)(40,55,56,54)(41,57,58,48)(44,61,62,60)(53,63,64,59)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=i,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^b=eh,e^c=eh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6262292);