Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #23185 |
| prime radical | #33716 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1772 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,15,8,14,26,48,9,16,25,11,6,33,4,13)(3,17,22,40,21,47,20,37,23,35,18,10,34,38,50,39)(5,12,30,43,31,45,28,58,32,46,54,41,29,59,24,44)(19,42,51,63,52,64,27,55,53,60,49,36,56,61,57,62);(1,3,8,34,9,23,6,21)(2,10,14,37,16,40,33,39)(4,18,7,20,26,22,25,50)(5,27,31,51,32,57,29,49)(11,35,13,47,15,17,48,38)(12,42,45,61,46,60,59,64)(19,30,56,28,53,54,52,24)(36,43,55,58,63,41,62,44);(1,4,6,25,9,26,8,7)(2,11,14,13,16,15,33,48)(3,18,21,50,23,22,34,20)(5,28,29,30,32,24,31,54)(10,35,37,47,40,17,39,38)(12,43,45,58,46,41,59,44)(19,51,52,27,53,49,56,57)(36,42,55,61,63,60,62,64);(1,5,7,30,8,31,26,28,9,32,25,54,6,29,4,24)(2,12,15,43,14,45,48,58,16,46,11,41,33,59,13,44)(3,19,18,49,21,52,50,57,23,53,22,51,34,56,20,27)(10,36,47,64,39,62,35,60,40,63,38,61,37,55,17,42)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=eh,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,e^b=eh,g^b=gh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405766);