Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479149 |
| 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:
(1,2,7,12,8,14,6,13)(3,15,20,34,19,41,18,9)(4,21,25,10,26,36,24,37)(5,27,31,39,30,40,29,11)(16,44,52,53,22,32,46,42)(17,47,50,43,51,33,49,57)(23,35,56,38,55,58,28,59)(45,63,60,61,54,62,48,64);(1,3,6,18,8,19,7,20)(2,9,12,15,14,34,13,41)(4,22,24,52,26,16,25,46)(5,17,29,49,30,51,31,50)(10,32,36,42,37,44,21,53)(11,33,27,57,39,47,40,43)(23,54,28,60,55,45,56,48)(35,61,38,62,58,64,59,63);(1,4,6,24,8,26,7,25)(2,10,12,36,14,37,13,21)(3,16,18,46,19,22,20,52)(5,28,29,55,30,56,31,23)(9,32,15,42,34,44,41,53)(11,38,27,58,39,59,40,35)(17,48,49,54,51,60,50,45)(33,62,57,64,47,63,43,61);(1,5,7,29,8,30,6,31)(2,11,13,27,14,39,12,40)(3,17,18,50,19,51,20,49)(4,23,25,28,26,55,24,56)(9,33,15,43,34,47,41,57)(10,35,21,38,37,58,36,59)(16,45,46,60,22,54,52,48)(32,61,42,63,44,64,53,62)Polycyclic presentation:
2^9|a^2=eg,b^2=e,c^2=e,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,h^c=hi,e^d=ei47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386345);