Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449963 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15954 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,28)(3,14,18,34,17,9,29,35)(4,19,22,38,23,10,30,39)(5,24,27,42,26,41,31,11)(15,43,46,50,45,32,20,36)(16,47,49,33,25,58,48,40)(21,53,57,37,55,63,56,51)(44,61,52,60,59,62,54,64);(1,3)(2,9,12,14)(4,20)(5,25,26,16)(6,29)(7,18)(8,17)(10,36,19,50)(11,40)(13,35,28,34)(15,22)(21,54,55,52)(23,46)(24,47)(27,48,31,49)(30,45)(32,39,43,38)(33,42)(37,62)(41,58)(44,56,59,57)(51,61)(53,60)(63,64);(1,4)(2,10)(3,15,17,45)(5,21)(6,22)(7,30)(8,23)(9,32,14,43)(11,37)(12,19)(13,38)(16,44,25,59)(18,20,29,46)(24,53)(26,55)(27,56)(28,39)(31,57)(33,60,40,64)(34,36,35,50)(41,63)(42,51)(47,62,58,61)(48,52,49,54);(1,5,8,26)(2,11,12,42)(3,16,17,25)(4,21,23,55)(6,27,7,31)(9,33,14,40)(10,37,19,51)(13,41,28,24)(15,44,45,59)(18,48,29,49)(20,52,46,54)(22,56,30,57)(32,60,43,64)(34,58,35,47)(36,61,50,62)(38,63,39,53)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=e,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357160);