Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6351869 |
| 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 | #3444 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
2545 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)(3,13,15,24)(4,18,21,25)(5,11,6,10)(8,16,23,17)(9,19,26,20)(14,22,28,32)(27,29,31,30)(33,34,39,42)(35,45,47,58)(36,41,51,55)(37,44,38,43)(40,52,54,48)(46,60,61,59)(49,57,50,56)(53,63,64,62);(1,3)(2,8)(4,14)(5,16)(6,17)(7,15)(9,22)(10,24)(11,13)(12,23)(18,31)(19,29)(20,30)(21,28)(25,27)(26,32)(33,35,39,47)(34,40,42,54)(36,46,51,61)(37,48,38,52)(41,53,55,64)(43,45,44,58)(49,62,50,63)(56,60,57,59);(1,4,7,21)(2,9,12,26)(3,14,15,28)(5,20,6,19)(8,22,23,32)(10,18,11,25)(13,27,24,31)(16,30,17,29)(33,36,39,51)(34,41,42,55)(35,46,47,61)(37,50,38,49)(40,53,54,64)(43,57,44,56)(45,59,58,60)(48,63,52,62);(3,15)(8,23)(13,24)(14,28)(16,17)(22,32)(27,31)(29,30)(34,42)(35,47)(41,55)(43,44)(46,61)(48,52)(56,57)(62,63);(1,5,7,6)(2,10,12,11)(3,16,15,17)(4,19,21,20)(8,24,23,13)(9,25,26,18)(14,29,28,30)(22,27,32,31)(33,37,39,38)(34,43,42,44)(35,48,47,52)(36,49,51,50)(40,45,54,58)(41,56,55,57)(46,62,61,63)(53,60,64,59)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,e^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=ghi,d^b=dh,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=ghi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6259074);