Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478598 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17156 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,9,18,34,19,35,17,44)(4,20,24,10,23,39,31,38)(5,25,28,11,30,42,29,43)(15,36,21,52,46,58,47,32)(16,40,49,60,50,61,26,33)(22,41,54,63,27,37,55,51)(45,59,56,62,48,57,53,64);(1,3)(2,9)(4,21,23,47)(5,26)(6,18)(7,17)(8,19)(10,36,38,58)(11,40)(12,44)(13,34)(14,35)(15,24,46,31)(16,28)(20,32,39,52)(22,45,27,48)(25,33)(29,50)(30,49)(37,59,41,57)(42,60)(43,61)(51,62,63,64)(53,54,56,55);(1,4,6,24,8,23,7,31)(2,10,12,20,14,38,13,39)(3,15,17,47,19,46,18,21)(5,27,28,55,30,22,29,54)(9,32,34,36,35,52,44,58)(11,41,25,51,43,37,42,63)(16,48,26,56,50,45,49,53)(33,59,40,62,60,57,61,64);(1,5,8,30)(2,11,14,43)(3,16,19,50)(4,22,23,27)(6,29,7,28)(9,33,35,60)(10,37,38,41)(12,42,13,25)(15,45,46,48)(17,49,18,26)(20,51,39,63)(21,53,47,56)(24,55,31,54)(32,57,52,59)(34,61,44,40)(36,62,58,64)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385795);