Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #479464 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17208 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,14)(3,15,18,34,20,9,19,43)(4,21,25,10,24,37,26,38)(5,27,30,42,29,41,31,11)(16,46,49,32,48,35,22,44)(17,50,28,33,51,39,52,45)(23,36,57,53,58,40,56,59)(47,60,54,63,55,61,64,62);(1,3,6,18,8,20,7,19)(2,9,14,43,12,15,13,34)(4,22,26,16,24,49,25,48)(5,28,31,51,29,52,30,17)(10,35,21,44,38,46,37,32)(11,39,41,45,42,50,27,33)(23,55,56,64,58,47,57,54)(36,60,59,63,40,61,53,62);(1,4,6,26,8,24,7,25)(2,10,14,21,12,38,13,37)(3,16,18,49,20,48,19,22)(5,23,31,56,29,58,30,57)(9,32,43,35,15,44,34,46)(11,40,41,53,42,36,27,59)(17,47,28,54,51,55,52,64)(33,61,39,62,45,60,50,63);(1,5,7,31,8,29,6,30)(2,11,14,27,12,42,13,41)(3,17,19,28,20,51,18,52)(4,23,26,57,24,58,25,56)(9,33,43,50,15,45,34,39)(10,36,37,59,38,40,21,53)(16,47,49,64,48,55,22,54)(32,60,46,63,44,61,35,62)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,c^2=ef,d^2=fi,f^2=i,g^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,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, 386661);