Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479151 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17188 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,30)(3,14,18,35,19,34,17,9)(4,20,31,37,24,53,23,10)(5,11,28,40,29,42,27,41)(15,32,47,43,21,54,46,60)(16,48,51,38,25,44,50,33)(22,39,56,63,57,52,26,36)(45,59,58,61,55,62,49,64);(1,3,6,17,7,19,8,18)(2,9,12,14,13,35,30,34)(4,21,23,47,24,15,31,46)(5,25,27,51,29,16,28,50)(10,32,20,43,37,54,53,60)(11,38,40,44,42,33,41,48)(22,45,26,49,57,55,56,58)(36,62,39,64,63,59,52,61);(1,4,6,23,7,24,8,31)(2,10,12,20,13,37,30,53)(3,15,17,46,19,21,18,47)(5,26,27,57,29,56,28,22)(9,32,14,43,35,54,34,60)(11,39,40,63,42,52,41,36)(16,49,50,55,25,58,51,45)(33,61,48,62,38,64,44,59);(1,5,6,28,7,29,8,27)(2,11,12,41,13,42,30,40)(3,16,18,50,19,25,17,51)(4,22,23,56,24,57,31,26)(9,33,34,48,35,38,14,44)(10,36,20,52,37,63,53,39)(15,45,47,49,21,55,46,58)(32,59,60,61,54,62,43,64)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=e,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,h^c=hi,e^d=ei,h^d=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386347);