Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #27987 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2504 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2504 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,14,7,13,6,12)(3,15,18,32,19,9,26,31)(4,20,27,36,23,35,22,10)(5,11,28,39,25,40,24,38)(16,41,43,33,44,54,50,29)(17,37,45,56,46,30,51,55)(21,47,52,59,49,58,48,34)(42,60,61,57,62,63,64,53);(1,3,6,18,7,19,8,26)(2,9,12,31,13,15,14,32)(4,16,27,50,23,44,22,43)(5,17,28,51,25,46,24,45)(10,33,20,41,36,29,35,54)(11,37,39,55,40,30,38,56)(21,42,48,61,49,62,52,64)(34,53,58,60,59,57,47,63);(1,4,6,22,7,23,8,27)(2,10,14,20,13,36,12,35)(3,16,18,43,19,44,26,50)(5,21,24,48,25,49,28,52)(9,29,32,54,15,33,31,41)(11,34,39,47,40,59,38,58)(17,42,45,61,46,62,51,64)(30,53,56,63,37,57,55,60);(1,5)(2,11)(3,17)(4,21)(6,24)(7,25)(8,28)(9,30)(10,34)(12,38)(13,40)(14,39)(15,37)(16,42)(18,45)(19,46)(20,47)(22,48)(23,49)(26,51)(27,52)(29,53)(31,55)(32,56)(33,57)(35,58)(36,59)(41,60)(43,61)(44,62)(50,64)(54,63);(1,6,7,8)(2,12,13,14)(3,18,19,26)(4,22,23,27)(5,24,25,28)(9,31,15,32)(10,35,36,20)(11,38,40,39)(16,43,44,50)(17,45,46,51)(21,48,49,52)(29,41,33,54)(30,55,37,56)(34,58,59,47)(42,61,62,64)(53,60,57,63)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365729);