Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62945 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2617 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,30)(3,14,17,42)(4,20,21,39)(5,25,31,27)(6,12,7,13)(9,32,34,16)(10,24,37,23)(11,29,57,28)(15,44,45,64)(18,40,19,43)(22,38,52,62)(26,35,49,36)(33,59,60,53)(41,48,63,47)(46,50,58,61)(51,56,54,55);(1,3,6,18)(2,9,12,35)(4,15,23,47)(5,26,28,16)(7,19,8,17)(10,33,20,50)(11,40,25,14)(13,36,30,34)(21,45,24,48)(22,53,55,46)(27,42,57,43)(29,32,31,49)(37,60,39,61)(38,44,51,41)(52,59,56,58)(54,63,62,64);(1,2,7,13)(3,9,19,36)(4,10,24,39)(5,27,29,11)(6,12,8,30)(14,16,43,49)(15,33,48,61)(17,34,18,35)(20,21,37,23)(22,54,56,38)(25,28,57,31)(26,42,32,40)(41,46,64,59)(44,53,63,58)(45,60,47,50)(51,55,62,52);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,21)(9,33)(11,38)(12,20)(13,39)(14,41)(16,46)(17,45)(18,47)(19,48)(25,51)(26,53)(27,54)(28,55)(29,56)(30,37)(31,52)(32,58)(34,60)(35,50)(36,61)(40,44)(42,63)(43,64)(49,59)(57,62);(4,21)(10,37)(15,45)(20,39)(22,52)(23,24)(33,60)(38,62)(41,63)(44,64)(46,58)(47,48)(50,61)(51,54)(53,59)(55,56)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=i,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260477);