Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #443899 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15768 |
| 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 | |
712 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,13,8,14)(3,15,18,34,20,33,19,9)(4,10,23,37,25,39,24,38)(5,26,28,11,29,41,30,42)(16,43,45,31,46,35,21,50)(17,47,27,56,49,32,48,40)(22,52,53,36,54,61,55,62)(44,58,57,59,64,60,51,63);(1,3,8,18,7,20,6,19)(2,9,14,15,13,34,12,33)(4,21,23,46,25,45,24,16)(5,27,30,49,29,48,28,17)(10,35,37,31,39,43,38,50)(11,40,26,47,42,56,41,32)(22,44,53,51,54,64,55,57)(36,60,61,59,62,58,52,63);(1,4)(2,10,13,39)(3,16,20,46)(5,22)(6,23)(7,25)(8,24)(9,31)(11,36,42,62)(12,37,14,38)(15,35)(17,44,49,64)(18,45,19,21)(26,52,41,61)(27,57,48,51)(28,53)(29,54)(30,55)(32,58)(33,43)(34,50)(40,59)(47,60)(56,63);(1,5,6,28,7,29,8,30)(2,11,12,41,13,42,14,26)(3,17,18,27,20,49,19,48)(4,22,23,53,25,54,24,55)(9,32,15,40,34,47,33,56)(10,36,37,61,39,62,38,52)(16,44,45,57,46,64,21,51)(31,58,35,59,50,60,43,63)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=egi,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351096);