Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #442706 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15727 |
| 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 | |
807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,29,7,13,6,12)(3,14,28,9,18,39,17,34)(4,19,22,47,23,10,30,36)(5,24,31,38,27,11,26,54)(15,40,42,45,20,48,50,32)(16,43,56,59,44,37,25,33)(21,35,52,61,53,46,57,64)(41,62,55,58,49,60,51,63);(1,3)(2,9)(4,20,23,15)(5,25)(6,28)(7,18)(8,17)(10,32,19,45)(11,37)(12,39)(13,34)(14,29)(16,26)(21,51,53,55)(22,42,30,50)(24,43)(27,56)(31,44)(33,38)(35,60,46,62)(36,48,47,40)(41,52,49,57)(54,59)(58,64,63,61);(1,4,6,22,7,23,8,30)(2,10,12,36,13,19,29,47)(3,15,17,42,18,20,28,50)(5,21,26,52,27,53,31,57)(9,32,14,40,34,45,39,48)(11,35,38,61,24,46,54,64)(16,41,25,55,44,49,56,51)(33,58,37,60,59,63,43,62);(1,5)(2,11,13,24)(3,16,18,44)(4,21)(6,26)(7,27)(8,31)(9,33)(10,35,19,46)(12,38,29,54)(14,37)(15,41,20,49)(17,25,28,56)(22,52)(23,53)(30,57)(32,58)(34,59)(36,61,47,64)(39,43)(40,60)(42,55,50,51)(45,63)(48,62)Polycyclic presentation:
2^9|a^2=efg,b^2=fgi,c^2=e,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 349903);