Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477926 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17126 |
| 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 | |
817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,9,19,33,17,35,18,34)(4,20,23,10,25,43,24,38)(5,26,28,41,30,42,29,11)(15,44,51,58,46,36,21,31)(16,47,27,32,49,39,50,59)(22,40,54,52,55,37,56,61)(45,63,64,60,48,57,53,62);(1,3,7,19,8,17,6,18)(2,9,14,34,12,35,13,33)(4,21,24,15,25,51,23,46)(5,27,29,49,30,50,28,16)(10,36,43,58,38,44,20,31)(11,39,26,32,41,47,42,59)(22,45,56,64,55,48,54,53)(37,57,61,60,40,63,52,62);(1,4,6,23,8,25,7,24)(2,10,13,20,12,38,14,43)(3,15,18,21,17,46,19,51)(5,22,28,54,30,55,29,56)(9,31,33,44,35,58,34,36)(11,40,42,61,41,37,26,52)(16,48,50,64,49,45,27,53)(32,57,39,62,59,63,47,60);(1,5)(2,11,12,41)(3,16,17,49)(4,22)(6,28)(7,29)(8,30)(9,32)(10,37,38,40)(13,42,14,26)(15,45,46,48)(18,50,19,27)(20,52,43,61)(21,53,51,64)(23,54)(24,56)(25,55)(31,57)(33,39)(34,47)(35,59)(36,60)(44,62)(58,63)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385123);