Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #510115 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #31314 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#576 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #1764 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
567 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,7,9,29,30)(3,13,28,34,16,8,18,40)(4,19,23,35,21,49,51,10)(5,11,26,39,27,36,25,58)(14,31,47,33,24,41,17,42)(15,44,54,32,20,50,46,37)(22,48,57,61,55,64,56,38)(43,60,53,62,52,63,45,59);(1,3,6,28,7,16,29,18)(2,8,30,34,9,13,12,40)(4,20,51,54,21,15,23,46)(5,24,26,17,27,14,25,47)(10,37,19,44,35,32,49,50)(11,31,58,42,36,41,39,33)(22,43,56,45,55,52,57,53)(38,62,48,63,61,59,64,60);(2,9)(3,14)(5,25)(6,29)(8,31)(10,35)(11,39)(13,41)(15,43)(16,24)(17,28)(18,47)(20,52)(22,56)(23,51)(26,27)(32,59)(33,34)(36,58)(37,62)(38,48)(40,42)(44,60)(45,54)(46,53)(50,63)(55,57)(61,64);(1,4)(2,10)(3,15)(5,22)(6,23)(7,21)(8,32)(9,35)(11,38)(12,19)(13,37)(14,43)(16,20)(17,45)(18,46)(24,52)(25,56)(26,57)(27,55)(28,54)(29,51)(30,49)(31,59)(33,60)(34,44)(36,61)(39,48)(40,50)(41,62)(42,63)(47,53)(58,64)Polycyclic presentation:
2^9|a^2=eh,b^2=efh,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=ghi,h^c=hi,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 417311);