Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251967 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(2,8)(3,13)(5,19)(6,27)(9,32)(11,37)(12,31)(14,38)(15,39)(16,48)(17,23)(18,41)(20,29)(25,57)(26,35)(28,60)(30,34)(33,54)(36,58)(40,59)(42,55)(43,45)(44,53)(46,47)(49,63)(51,61)(52,56)(62,64);(1,2,7,12)(3,14,42,64)(4,8,21,31)(5,22,26,10)(6,28,29,60)(9,33,47,49)(11,27,37,20)(13,38,55,62)(15,16,30,59)(17,44,18,45)(19,50,35,24)(23,53,41,43)(25,58,57,56)(32,54,46,63)(34,40,39,48)(36,51,52,61);(1,3,7,18)(2,9,12,15)(4,17,21,42)(5,23,26,55)(6,16,29,33)(8,30,31,47)(10,32,22,39)(11,14,37,45)(13,35,41,19)(20,49,27,59)(24,34,50,46)(25,54,57,48)(28,44,60,64)(36,62,52,53)(38,56,43,58)(40,61,63,51);(1,2,7,12)(3,15,18,9)(4,8,21,31)(5,24,26,50)(6,11,29,37)(10,35,22,19)(13,39,41,32)(14,16,45,33)(17,47,42,30)(20,28,27,60)(23,46,55,34)(25,56,57,58)(36,61,52,51)(38,48,43,54)(40,53,63,62)(44,49,64,59)Polycyclic presentation:
2^9|b^2=ehi,c^2=i,d^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fh,f^d=fh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159189);