Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351101 |
| 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 | #62939 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,16,26)(4,9,17,23)(5,11,6,10)(8,20,22,15)(14,27,28,32)(18,25,19,24)(21,30,31,29)(33,34,39,45)(35,46,50,56)(36,42,53,58)(37,44,38,43)(40,48,57,49)(41,52,59,51)(47,61,64,54)(55,63,60,62);(1,3)(2,8)(4,14)(5,15)(6,20)(7,16)(9,21)(10,13)(11,26)(12,22)(17,28)(18,29)(19,30)(23,31)(24,27)(25,32)(33,35,39,50)(34,40,45,57)(36,47,53,64)(37,48,38,49)(41,54,59,61)(42,55,58,60)(43,56,44,46)(51,62,52,63);(1,4)(2,9)(3,14)(5,18)(6,19)(7,17)(8,21)(10,24)(11,25)(12,23)(13,27)(15,29)(16,28)(20,30)(22,31)(26,32)(33,36,39,53)(34,41,45,59)(35,47,50,64)(37,51,38,52)(40,54,57,61)(42,44,58,43)(46,60,56,55)(48,62,49,63);(4,17)(9,23)(14,28)(18,19)(21,31)(24,25)(27,32)(29,30)(33,36)(34,42)(35,47)(37,52)(38,51)(39,53)(40,55)(41,43)(44,59)(45,58)(46,61)(48,63)(49,62)(50,64)(54,56)(57,60);(1,5,7,6)(2,10,12,11)(3,15,16,20)(4,18,17,19)(8,13,22,26)(9,24,23,25)(14,29,28,30)(21,27,31,32)(33,37,39,38)(34,43,45,44)(35,48,50,49)(36,51,53,52)(40,56,57,46)(41,42,59,58)(47,62,64,63)(54,55,61,60)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6258506);