Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #502133 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #24280 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
731 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6,13,37,33,50,32,49,12,19,10,17,9,16,11,18)(7,14,38,62,59,66,23,42,24,43,22,41,21,40,35,52)(8,25,36,47,27,46,34,15,31,53,28,45,30,51,29,48)(20,54,60,64,55,68,26,39,58,67,56,63,57,44,61,65);(2,4)(5,7,10,22,33,59,11,35,12,24,13,38,9,21,32,23)(6,14,17,41,50,66,18,52,19,43,37,62,16,40,49,42)(8,26,28,20,27,56,29,55,31,61,36,58,30,60,34,57)(15,44,25,39,45,54,46,63,48,68,53,65,47,67,51,64);(5,8)(6,15)(7,20)(9,27)(10,34)(11,36)(12,31)(13,29)(14,39)(16,45)(17,51)(18,53)(19,48)(21,55)(22,26)(23,56)(24,58)(25,49)(28,32)(30,33)(35,60)(37,46)(38,61)(40,63)(41,44)(42,54)(43,65)(47,50)(52,67)(57,59)(62,68)(64,66);(2,4)(5,9,12,33)(6,16,19,50)(7,21,24,59)(8,27,31,30)(10,32,13,11)(14,40,43,66)(15,45,48,47)(17,49,37,18)(20,55,58,57)(22,23,38,35)(25,46,53,51)(26,56,61,60)(28,29,36,34)(39,63,65,64)(41,42,62,52)(44,54,68,67)Polycyclic presentation:
2^9|a^2=fi,b^2=f,d^2=i,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fh,g^c=ghi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 409334);