Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#578 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2)(3,4)(5,6,11,16)(7,17,21,40)(8,22,26,13)(9,14,27,38)(10,29,31,15)(12,30,34,20)(18,43,23,41)(19,42,46,62)(24,48,52,36)(25,37,54,49)(28,56,55,39)(32,45,35,51)(33,57,59,47)(44,64,50,63)(53,61,68,66)(58,65,60,67);(5,7,11,21)(6,12,16,34)(8,23,26,18)(9,19,27,46)(10,30,31,20)(13,35,22,32)(14,33,38,59)(15,40,29,17)(24,50,52,44)(25,45,54,51)(28,57,55,47)(36,60,48,58)(37,41,49,43)(39,62,56,42)(53,65,68,67)(61,63,66,64);(2,4)(5,8,10,25,11,26,31,54)(6,13,15,37,16,22,29,49)(7,18,20,45,21,23,30,51)(9,24,28,53,27,52,55,68)(12,32,17,41,34,35,40,43)(14,36,39,61,38,48,56,66)(19,44,47,65,46,50,57,67)(33,58,42,63,59,60,62,64);(5,9)(6,14)(7,19)(8,24)(10,28)(11,27)(12,33)(13,36)(15,39)(16,38)(17,42)(18,44)(20,47)(21,46)(22,48)(23,50)(25,53)(26,52)(29,56)(30,57)(31,55)(32,58)(34,59)(35,60)(37,61)(40,62)(41,63)(43,64)(45,65)(49,66)(51,67)(54,68);(9,27)(14,38)(19,46)(24,52)(28,55)(33,59)(36,48)(39,56)(42,62)(44,50)(47,57)(53,68)(58,60)(61,66)(63,64)(65,67)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,f^b=fi,g^c=gi,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339125);