Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #389324 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14539 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,30)(3,9,17,34,16,35,18,33)(4,19,24,38,22,49,23,10)(5,25,28,11,29,40,27,39)(14,41,43,31,44,36,20,50)(15,45,46,32,26,55,47,58)(21,37,54,61,52,62,53,48)(42,63,51,59,56,57,64,60);(1,3,6,17,8,16,7,18)(2,9,13,33,12,35,30,34)(4,20,23,14,22,43,24,44)(5,26,27,47,29,15,28,46)(10,36,19,31,38,41,49,50)(11,32,40,45,39,58,25,55)(21,51,53,56,52,64,54,42)(37,60,61,57,62,59,48,63);(1,4)(2,10,12,38)(3,14,16,44)(5,21)(6,23)(7,24)(8,22)(9,31)(11,37,39,62)(13,19,30,49)(15,42,26,56)(17,43,18,20)(25,48,40,61)(27,53)(28,54)(29,52)(32,57)(33,41)(34,36)(35,50)(45,59)(46,51,47,64)(55,60)(58,63);(1,5,8,29)(2,11,12,39)(3,15,16,26)(4,21,22,52)(6,28,7,27)(9,32,35,58)(10,37,38,62)(13,25,30,40)(14,42,44,56)(17,47,18,46)(19,48,49,61)(20,51,43,64)(23,54,24,53)(31,57,50,63)(33,55,34,45)(36,59,41,60)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^d=fi,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296523);