Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #478348 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2754 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #17227 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,28)(3,14,30,34,18,43,17,9)(4,10,29,36,22,37,21,41)(5,23,31,57,27,11,26,40)(15,42,50,32,19,51,45,60)(16,46,24,33,49,38,48,62)(20,53,58,56,55,35,25,39)(44,61,54,64,52,63,47,59);(1,3,8,30,7,18,6,17)(2,9,28,43,13,34,12,14)(4,19,21,50,22,15,29,45)(5,24,26,16,27,48,31,49)(10,32,36,51,37,60,41,42)(11,38,40,62,23,46,57,33)(20,54,58,52,55,47,25,44)(35,63,56,64,53,61,39,59);(1,4,7,22)(2,10)(3,15,18,19)(5,25,27,58)(6,29,8,21)(9,32)(11,39)(12,41)(13,37)(14,42)(16,47,49,54)(17,50,30,45)(20,31,55,26)(23,56)(24,52,48,44)(28,36)(33,61)(34,60)(35,40)(38,59)(43,51)(46,64)(53,57)(62,63);(1,5,6,26,7,27,8,31)(2,11,12,40,13,23,28,57)(3,16,17,48,18,49,30,24)(4,20,21,25,22,55,29,58)(9,33,14,38,34,62,43,46)(10,35,36,39,37,53,41,56)(15,44,45,47,19,52,50,54)(32,59,51,61,60,64,42,63)Polycyclic presentation:
2^9|a^2=eg,b^2=efg,c^2=g,d^2=e,e^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,d^c=de,e^c=ei,f^c=fi,f^d=fi,h^d=hi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386907);